How is possible for current to flow so fast when charge flows so slow? How is it possible for current to flow so fast when charge flows so slowly?
We know electrons travel very slowly while charge travels at ~the speed of light.
 A: What is being confused here is not the flow of "current" but rather the transmission of energy.
The individual electrons in a wire move very slowly, as they can be modeled as constantly colliding with atoms (yes, this is a naive classical model, no quantum) and bouncing around randomly in the manner of a gas (the term "electron gas" is real and not inappropriate at all). Electric current is the very slow flow of this electron gas through the wire when an electric field is present. The term "flow of current" actually is misleading - there is no such "substance" called "current", current is a flow. "Flow of charges" or more specifically (in this case - in others, it may differ!) "flow of electrons" makes more sense. (After all, we don't talk about "current" as a substance which is contained within a river and which is what does the "flowing", i.e. "flow of current in the river", rather we talk of flowing "water in" the river, and "the current" means the flow of water.) See:
http://amasci.com/miscon/eleca.html#cflow
Energy, however, is not transmitted by one electron moving all the way around the circuit to the load, but rather through waves in the electrons and more importantly, the associated electric field. It's the same way that mechanical energy is transmitted in, say, a pole that is pushed from one end. The pole compresses slightly, and a sound wave thus appears, initially containing all the energy within your "push", and then travels down it, progressively distributing that energy amongst all the atoms within the pole until they are all moving in a single direction (here I imagine the pole pushed in a vacuum, as in interstellar space, with no other forces acting). The same goes with electrons in the circuit - though I should point out the following model is a bit simplistic but is more to convey the point of how the energy is transmitted than to detail the actual behavior of the electrons, which involves quantum mechanics and is subject to many of the same caveats as one sees within in an individual atom or molecule. But in this loose sense, when you throw the switch, now an electromagnetic wave travels down, setting the electrons ahead in motion and thus distributing its energy throughout the circuit. Of course, the core atoms of the metal are relatively fixed despite the electron motion, so the latter will tend to lose that energy to collision with them, unlike the pole where everyone, atoms and electrons together, start going in synchrony, and thus you have to keep supplying energy to them with a power source like a battery or generator which effectively keeps "pushing the pole" and thus keeps energy going into it - now think about a pole that is now not in vacuum but in molasses, and you have to keep pushing it to keep it moving. This pushing on atoms, of course, is how electrical devices can use electrically transmitted energy to do useful tasks.
Electromagnetic waves, and sound waves, thus energy, travel much faster than the electrons and the atoms in both the circuit and pushed pole. Energy is what lights up your light bulbs, and energy is what makes your computer operate. Since energy travels fast, these devices start operating "at the flick of a switch".
A: One needs to distinguish between two things when it comes to electricity, electric currents and voltages. 
1) The electric current is flow of electrons in metal wires, (or in fluids like electrolytes). The electrons are moving in the wire at the drift velocity
$v=\frac{I}{enA}$
where: $I$ is the electric current; $e$ is the elctric charge on the electron; $n$ is the electron number density in the metal material of the wire; $A$ the cross section area of the wire.
Depending on the values of $I$, $n$ and $A$, the speed $v$ has a typical value of several cms$^{-1}$! 
2) However, the cause of the motion of the electrons is the electric field, that you set in the wire when you switch on the light say, that travells along the wire at the speed of light. As the field travells along the wire so fast, it sets the electrons along the way into motion all along the wire. So it appears as if the electrons are moving very fast, when in fact they don't. I hope this clarifies your point you were trying to make?
A: As you may've noticed, the drift velocity of electric charges is $\sim {10}^{-5} \frac{\mathrm{m}}{\mathrm{s}} ,$ which is much slower than the flow of electric current.
So, how could electric current travel near the speed of light?  It's because the charges' electric field is propagating near the speed of light.
A: It seems you are contrasting the speed of propagation of current with the speed of the individual charge carriers.
These two things are clearly separate.  There are many examples.  Consider sound.
A fire cracker goes off at the other end of a football field from you.  You hear the sound a few 100 ms later.  The air molecules that were by the firecracker didn't end up by you.  They didn't travel far at all.  However, they pushed on their neighbors, which pushed on their neighbors, etc, all the way to your ears.  This pushing can propagate a lot faster than individual molecules can move.
Think of a long hollow cardboard tube filled with small balls just a little smaller than the inside diameter of the tube.  All the balls are touching each other.  You push on one ball on one end and move it 1 mm.  The ball at the other end then moves 1 mm.  However, none of the balls themselves moved more than 1 mm and they did that as slowly as you pushed, yet the propagation of the push was instantaneous on your human scale.
A: The current is measured in Amperes, which are the number of Coulombs of electric charge passing a fixed cross-sectional area per unit time. It says nothing about the speed of the electrons. Usually when we produce a current with an electric field on a conductor, it is a static electric field, so the speed of propagation of changes in the electric field isn’t an issue. The electric field causing the current is assumed to have always been (and will always be) the same.
A: The moving charged particles within a conductor move constantly in random directions, like the particles of a gas. In order for there to be a net flow of charge, the particles must also move together with an average drift rate.That's what drift velocity is called.The drift velocity deals with the average velocity that a particle, such as an electron, attains due to an electric field. In general, an electron will move in a conductor randomly. Free electrons in a conductor vibrate randomly, but without the presence of an electric field there is no net velocity. When a DC voltage is applied the electrons will increase in speed proportional to the strength of the electric field. These speeds are on the order of millimeters per hour. AC voltages cause no net movement; the electrons oscillate back and forth in response to the alternating electric field Electrons are the charge carriers in metals and they follow a random path, bouncing from atom to atom, but generally drifting in the opposite direction of the electric field. The speed at which they drift can be calculated from the equation:
I=nAvQ
where
I is the electric current
n is number of charged particles per unit volume (or charge carrier density)
A is the cross-sectional area of the conductor
v is the drift velocity, and
Q is the charge on each particle.
Similarly when we tak about the speed of electricity we generally talk about the flow of signal not about the velocity of electrons.
