Is it possible to play car racing games with a joystick?

Of course it is possible.

The game doesn't care what you use for inputs. It'll simply read your button presses and control the car as such.

People have played Dark Souls using music instruments, so driving with a joystick, even if it is a fancy flight stick, isn't out of the question.

Of course, if you're asking whether it is the best experience, that's up to you to determine.

This answer assumes you talk about a flight simulator standard joystick, and not a gamepad stick (although that one would also work the same way).

It is possible, and I have done it. You will have to properly set-up the axes in your game's setting, but for left-right control it is a good alternative.

In my specific case, I tried two methods to map the throttle to the joystick with mixed results:

1. Map to the vertical axis: It was uncomfortable, and turns would command undesired acceleration.
2. Map to the joystick thrust handle: It worked pretty well, although my specific joystick set-up made it uncomfortable. I set a threshold for the acceleration to turn into reverse, as you would do for jet TOGA/AB and it was good enough.

Disclaimer: I'm not into racing games anymore, so I don't know how the setting screens look for recent ones. Just try to check if your game of choice accepts any random input method, which they should.