A doubt in the concept of potential energy The concept of potential energy is best understood with examples, so let me give an example:  
There lies a mass of $1~\mathrm{kg}$ on ground, I lift the mass to a height of $1~\mathrm m$. Since I have done a work (force times displacement) against the gravity, therefore, I have given potential energy to the mass. 
Well, how can I know that the mass has gained any energy, if you say that “Let it fall and it will gain kinetic energy”. This argument shows that potential energy that I gave to the mass during the lift is now converting into kinetic energy, but I see it as the gravity doing work on the mass and therefore its kinetic energy is increasing.  
Gravity is a force, so it can provide kinetic energy to a body by setting it in a motion and that’s exactly what’s happening in the above argument. Just imagine that I lifted that mass of $1~\mathrm{kg}$ and at when I reached to a height of $1~\mathrm m$ I just switched off the gravity. Now, will the potential energy convert into kinetic energy if I let the mass to fall (the word fall here refers only to my leaving it there). I think no, it will just remain there. So, it can be argued that energy-conservation is violated and hence the thought experiment is sloppy, but this contradiction of energy-conservation has come only from axiom of potential energy (the axiom that working against a field lets the object to gain some kind of energy called potential energy) So, why there is a concept of potential energy when it can be understood as the field is doing the work and supplying the kinetic energy rather than potential energy is getting converted to kinetic energy?  
A Second View of the Situation: Let's take magnetic field  instead of gravitational field, imagine that I have two wires with current flowing in both of them in same direction, like this

According to the principles of magnetism, the two wires will attract each other, now if I take the violet wire far away (all I mean is doing the work against the attractive field) and then suddenly switching off the current in the red wire, would my violet wire accelerate towards the red one? If not where the potential energy has gone which I supplied it during taking it away.  
We can also imagine this situation using electrostatic field (as @FGSUZ has said in his comment) by using parallel plate capacitors. We can do same thing in this setup too.   
Thank you. Any help will be much appreciated.  
Explanation of why I don’t think that my question is a duplicate: The question which was suggested to me as of which mine is a duplicate, I think that question is basically on work-energy theorem, there OP’s main problem was why potential energy came when work is defined as change in kinetic energy. My question is why do we say that potential energy converts into kinetic energy when it is the force(the gravitational force) which is supplying the kinetic energy? 
Another defense against duplication: The question, to which my question is seemed to be of duplicate, says clearly in the last second paragraph above the EDIT as 
"Secondly, the force done by us (the external agency), is along the direction of the displacement, the work done is positive and so the change in kinetic energy should be positive (by the work energy theorem?)"
It says clearly that the OP wants to understand things in terms of energy theorem and doubts within it. My question deals with the interpretation of conversion of PE into KE and the work done by the field to supply KE.  
 A: I prefer slightly different point of view on this problem.
We have some field and some object. There is some force acting on the object in the field. In some situations the nature of the force is such that:


*

*the force depends on position of the object only

*when we move the object from some arbitrary point A to arbitrary point B the work done by this force depends only on points, and does not depend on the path


In this case we can describe the field not by specifying the force at each point, but by specifying some scalar value at each point - the energy.
Mathematically these descriptions of the field are the same: if we know the energy we can calculate the forces at each point, if we know forces - we can arbitrarily select a value of energy in some point and calculate energy in all other points.
This approach usually is very convenient, but not all the forces can be described by potential energy. F.e. friction force - the work done by friction force depends on path we move object. Another example is magnetic field - the force acting on a charge in magnetic field depends on velocity of the charge.

So, why there is a concept of potential energy when it can be
  understood as the field is doing the work and supplying the kinetic
  energy rather than potential energy is getting converted to kinetic
  energy?

Couple of notes here.
It's not necessary that potential energy is converted to kinetic energy. It may be converted to other kinds of potential energy (electrostatic to gravitational for example).
Description of field in terms of energy is often much more convenient than the description in terms of forces. I guess this is the main reason why the concept of potential energy exists. And the deeper into hardcore physics - the less convenient forces description become.
Let me take out part of your question: "... potential energy ... can be understood as the field is doing the work...". This is absolutely correct :)
A: Your question has a few problems.
The definition of potential energy as the work done to move the object against the force is not a very good definition (although it is often repeated).  A better definition is potential energy is the negative of the work done by the internal (conservative) forces $$ \Delta U = -W_\mathrm{internal}$$
As pointed out by @Farcher a single object cannot have potential energy.  By definition, it is necessary to have a conservative force acting between the objects, and the energy is stored in the system comprising the two objects.
However, the most important problem with your scenario is this: your hypothetical scenario is not physically possible.   How can we expect our physics, which is based on actual observations of nature, to make predictions of unphysical behavior?   "Garbage in, garbage out."
A: 
So, why there is a concept of potential energy when it can be understood as the field is doing the work and supplying the kinetic energy rather than potential energy is getting converted to kinetic energy?

It is perfectly possible to describe the situation in term of work done by the field supplying kinetic energy. That's the content of work-energy theorem. However, the concept of potential energy is very useful, in the case it can be introduced (work of conservative forces).
Reasoning in term of potential energy is useful for different reasons: it is independent on the actual dynamics (it does not require to solve equations of motion), it allows to separate contributions coming fom different forces (work-energy theorem establishes a connection between total force and kinetic energy) and simplifies the analysis of complex dynamical situations by using a conservation principle.
Now, what about your switching off the gravitational field? Notwithstanding its apparent unphysical set-up, if referred to gravitational forces, I think that Farcher's  answer contains the right information, maybe not in the clearest form.
It is clear that the same problem (how conservation of energy can work if the field is suddenly switched off? in the presence of the field there was some potential energy which seems to disappear  when the field is switched off) is present in other contexts where it is easier to see where the energy goes. For instance, a charge in the presence of a charged body. 
Bringing a finite charge at a finite distance from the body implies some work of the electrostatic force, therefore there will be a potential energy.  That potential energy is a property of the whole system (added charge + charged body). In order to switch off the field, one has to bring all the charges from the charged body to infinity. Thus, due to the presence of the added charge, there will be a work exactly compensating  the potential energy difference. 
The same holds for any other way of "switching off" fields. Switching off is cannot be realized without a process where the variation of potential energy of one subsystem is not recovered by some other part of the system. It's the conservation of energy.
A: Your overall question is largely a pedagogical rather than physics one: you don't understand the concept of potential energy, and with regard to that issue, we can only make guesses as to what explanations will make things click for you. You are making a distinction between "the field is doing the work and supplying the kinetic energy" versus "potential energy is getting converted to kinetic energy", but it's not clear what you think the distinction is. That energy is stored in the field, and can be released into kinetic energy, is what potential energy means. For simplicity sake, lower level physics courses speak of the weight having a potential energy, but that is just a simplification. It's the weight-Earth system that has gravitational potential energy.
But for your wire thought experiment, there is a physics issue to be addressed. You seem to think that if you turn off the EMF powering a circuit, the current instantly disappears. But that is not how electricity works. There is energy stored in the current itself, and the current will continue flowing until that energy dissipates. Each circuit has a quantity called "inductance" that measures how much energy per unit of current is stored in the circuit. Moving the wires apart increases their inductance, and thus either the current in the wires will decrease, or the energy stored in the circuit will increase. When you cut the power, this energy will be released.
A: 
So, why there is a concept of potential energy when it can be
  understood as the field is doing the work and supplying the kinetic
  energy rather than potential energy is getting converted to kinetic
  energy?

Work (and heat) is energy transfer. Work does not create energy. It transfers energy. That includes the work done by gravity.
When something does positive work on something else there is a transfer of energy from that something to that something else.  When gravity does positive work on the falling object it doesn’t create the kinetic energy of the object. It transfers the gravitational potential energy stored the Earth/object  system to the object in the form of kinetic. 
Likewise the potential energy of the Earth/object system was not created by the gravitational field. You, an external agent did positive work on the object in raising it. Energy (chemical potential) was transferred from your body to the object. Simultaneously gravity, with a force acting opposite to the direction of the displacement of the object, does an equal amount of negative work taking the energy you gave the object and storing it as gravitational potential energy of the Earth/object system. Strictly speaking, gravity is not a source of energy. Energy can be stored and retrieved from the gravitational field.
Bottom line: The source of the kinetic energy of the falling object is the external agent that raised the object in the gravitational field.
UPDATE:
Based on our follow up discussions in Chat, it appears the main issue you are having with my answer is whether gravitational potential energy (GPE), or for that matter any form of potential energy (PE), is a property of an object alone or a system property. I submit to you that it is a system property and not a property of an object alone.
Although many times in physics discussions we talk about an object having GPE or PE, it is really for convenience rather than alway saying, for example, that GPE is possessed by the "object-Earth system", or that electrical potential energy of a charge is possessed by the "charge-electric field system", or that the potential energy of a mass attached to a massless spring stretched or compressed is possessed by the "mass-spring system". In fact, however, any form of potential energy is a system property. Objects by themselves do not "possess" PE. As pointed out in the following and many other websites:
"Potential energy is a property of a system rather than of a single object—due to its physical position". https://courses.lumenlearning.com/physics/chapter/7-3-gravitational-potential-energy/
So what it boils down to is this. Is potential energy is a property of an object alone, or is it a property of a system? It is my firm belief it is the latter.
Hope this helps.
A: Your statement  

I have given [gravitational] potential energy to the mass. 

is incorrect.  
It is the mass and Earth system which gains the gravitational potential energy not just the mass.
When the separation between the mass and the Earth decreases gravitational potential energy is converted to kinetic energy.
How then might you switch off gravity?
By moving the Earth an infinite distance from the mass.
To do this external work needs to be done on the mass and Earth system giving the system even more gravitational potential energy.  
If you are considering just the mass alone as the system then the external force acting on the mass is the attractive gravitational force due to the Earth.
Another external force needs to be applied on the mass to move it away from the Earth.
If the mass starts and finishes with no kinetic energy then no net work will have been done on the mass.
Remove the “other” external force and the attractive gravitational force on the mass due to the Earth does work on the mass and the kinetic energy of the mass increases.
“Switching off gravity” does nothing to the mass except removing the external force acting on it.
With no external force on the mass to do work on the mass the kinetic energy of the mass does not change.
A: Historically, potential energy was introduced because it is a useful concept; the other answers have already covered this aspect.
However, it is also worth pointing out that potential energy has real and measurable consequences that your alternative model cannot explain.  For example, the negative potential energy in an atomic nucleus directly affects the mass of the atom.  According to Wikipedia, this is referred to as the mass defect and it would be difficult to explain without using the concept of potential energy.
(Now, in your particular example, things get a bit tricky because of General Relativity, which I don't understand well enough to discuss here.  But I believe it is still true that the interaction between the 1kg mass and the Earth - which in classical mechanics we would represent as a potential energy - contributes to the overall gravitational field of the system as a whole.)
A: I really wanted to make this a comment on lesnik's insightful answer, but I don't have enough reputation to do that, so I am just going to elaborate on my thought.
When you say "why do we say that potential energy converts into kinetic energy when it is the force(the gravitational force) which is supplying the kinetic energy?" I think you are noticing the relation between forces that come from fields and potential energy. For any (conservative) force (like gravity), we can define a potential energy associated with that force.
Putting this relation in math, we relate the potential energy $U$ to the force $F$ by:
$$ F = - \frac{dU}{dx}$$
Or, with the more general (and spicier) notation:
$$ F = - \nabla U$$
The important takeaway is that potential energy is a definition. Potential energy is the special quantity whose derivative gives us the force $F$.
We can also move in the backwards direction using the same equation, for any potential energy, we can define a force $F$.
Your question boils down to two perspectives of the same thing: whenever potential energy changes, we can define a force to quantify this change. And conversely, whenever we have a (conservative) force, we can define a potential energy that tracks the change in kinetic energy (work) done by the force.
A: One definition of energy is the capability to perform work. But it only applies to entire systems and interactions therein (energy is only useful because there’s the conservation law, and that one only applies to systems).

According to the principles of magnetism, the two wires will attract each other, now if I take the violet wire far away (all I mean is doing the work against the attractive field) and then suddenly switching off the current in the red wire, would my violet wire accelerate towards the red one? If not where the potential energy has gone which I supplied it during taking it away.

Let’s be specific. You start with two coils of isolated wire at a certain distance from each other, and two batteries (unconnected). Suppose the coils are initially held in place by clamps of some kind. The batteries have the capability of performing work, the coils do not.
Now you connect the batteries to the wires. There is now an attractive force, and if not for the clamps, the coils would start moving. They are now eager to spend the batteries’ energy to start moving, and if friction would impede the movement, they would perform work against that friction. Thus the coils have gained energy, but not from nowhere.
Then you unclamp the wires and let them get closer to each other, let’s say all the way until the wires touch. This has caused the batteries to discharge a bit to give kinetic energy to the coils, and on impact (suppose it was inelastic) the energy dissipated as heat.
Now you clamp one coil in place and use an electric winch to pull the other one far away. The winch expends the energy of whatever power source it’s plugged into to do so. While the coil is moving, its movement through a magnetic field causes currents to flow through both wires, charging both batteries.
To answer your questions:


*

*If you move the wires sufficiently far away from each other, the change in the current in one of them isn’t going to affect the other in a noticeable way.

*The energy caused movement, movement caused magnetic flux change, which caused an electromotive force, which caused additional currents, which charged the battery and/or heated the wire.

A: 
Just imagine that, I lifted that mass of 1 kg and at when I reached to a height of 1 m I just switched off the gravity.

Your thought experiment assumes that you can "switch of gravity" without extracting or investing energy in it. The gravitational field has, as other fields, an energy density associated with it. I would pretty much bet that when you calculate the difference in field energy of the two configurations you mentioned, it will match the difference which you are missing.
A: There's no need to call into question relativity, in this context the old Newton law is still valid. Even if it is an approximation the force is applied to the center of gravity of both objects. So if you increase the distance between your sample mass and the center of the earth you also increase the distance along which it could keep accelerating if it could fall to the center of the earth. In the case of the wires if you move one of them while the current is flowing by moving some charges within a magnetic field you create another electromagnetic wave, so the extra energy is radiated away.
A: 
just switched off the gravity. Now, will the potential energy convert into kinetic energy if I let the mass to fall (the word fall here refers only to my leaving it there). I think no, it will just remain there.

Your assumption is a clear contradiction of Newton's First Law of Motion, which states as:
That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it.
If you would turn off the gravity, the mass will remain moving and that motion will not involve potential energy since it is field dependent.The velocity (& K.E.) of the mass will be exactly same as it was last time when the field was on (if there is no other force working against like drag etc.).
That's how spacecrafts move in the space ,once they reach a target velocity the thrusters are turned off and craft sails through the space with (nearly)same velocity since there is no air drag etc.
So it is not like field pulling as if it were ,the mass would have stopped immediately.
