Car on circular track problem 
A car starts from rest on a horizontal circular road of $190\ \mathrm m$ and gains speed at a uniform rate of $1.2\ \mathrm{m/s^2}$. The coefficient of static friction between the tyres and the road is $0.37$. Calculate the distance travelled by the car before it begins to skid.
Here is my doubt regarding the solution and the problem statement too.
As the solution given in the picture above, we can see that they have considered the centripetal force and the tangential force. But if I observe from the frame of the car,then centrifugal force applies for me. Then the direction of the net force is becoming different,so is the solution wrong?
Also this is kind of contradictory because if that happens there is no resultant force in the inward direction and so there is no force to balance the centrifugal force,so this is a huge dilemma at hand.So how will the net force be defined in this system?
Is this free body diagram even correct? I look forward to the brilliant answers of all the physics lovers out there to this question.
 A: 
if that happens there is no resultant force in the inward direction

In the frame of the car, the car does not accelerate.  So we would expect the net forces to be zero.
A: 
But if I observe from the frame of the car,then centrifugal force applies for me.

Yes, this is true!

Then the direction of the net force is becoming different

True; in the frame moving with the car the acceleration of the car is $0$. So the net force is $0$ if we want to bring in pseudo-forces to keep Newton's second law valid.

Also this is kind of contradictory because if that happens there is no resultant force in the inward direction and so there is no force to balance the centrifugal force

Here is the common misconception. The centripetal force (friction here) doesn't go away in the rotating frame! It's a real force, and so it exists in all frames of reference. However, in the frame rotating with the car, the centrifugal force exactly cancels the centripetal force.
If the frame is accelerating with the tangential acceleration of the car as well, then there is a tangential pseudo-force which cancels the tangential force acting on the car as well. This results in no acceleration of the car in the frame moving with the car.
If the frame is rotating at some fixed rate, then we will still have a centripetal and centrifugal force, but they will not always be equal, and so we will still observe circular motion. This makes sense, because a frame not rotating at the rate of the car at all times will see the car moving in a circular path. In this fixed rotation frame, the previously mentioned tangential pseudo-force would no longer be present.
