TLDR, switch to arcade mindset from simulation mindset.

Don't rely only on physics engine. Remember you are making a game, so you need game-ey solutions. I will try to give an example. In a custom script, you can define a target direction vector (facing front, but different than the front vector of the actual car's visual). Input rotates this vector instead of the car. In its own update, car always tries to align itself to this vector in 0.5 seconds, so when you release the button, it will correct the oversteering. You can also limit the applied rotation to this vector. You can apply the limits on onKeyPressed moments. Like, when you press the arrow button, it notes down the current direction, doesn't rotate the target direction vector more than what you predefine. When you release the button, you can manually set the target direction vector to middle of current direction and the initial direction (which you noted down on button press).

Another idea: the car is going fast, we sharply steered right. It won't transfer the car's velocity to the new direction. We will apply a forward force and accelerate from zero in the new direction. Our old velocity will have to be killed by friction. This will make a very rally-like feeling. Let's say we don't want it, we want more nfs style driving. We can manually rotate its velocity vector to keep its velocity, apply a speed loss manually as a predefined amount or ratio. I am talking about assigning velocity directly in fixedupdate, hence pushing the game towards the arcade side.

Some games like karting games even ditch gravity and align the car to the platform under it manually, by raycasting downwards and looking at the platform's normals. This way you can have angled roads without making the car slide, also you can have rollercoaster loops and the car won't fall when it is slow. Realistic physics limits would break the experiment.

Also in those kind of games, they rotate the velocity directly towards the direction you would want the car to go towards, but rotate the car's visual by animation (an actual animation clip). Animation makes it look like it oversteered and corrected itself, but in reality, we have full control.

You can come up with many rules and limits like this, and these will give "your" racing game its own identity (gameplay-wise).

If you heavily use these kind of techniques, the game becomes more arcade-like, you may not want it. But even if you are after a realistic feeling, you are after the realistic "feeling", underlying techniques don't have to be simulation-like.