The 'trick' is to treat it like a real car. You really need to balance your steering with accel/brake. The hardest part after this is pre-empting what the car in front will do. My strategy is if I fail to qualify in 3rd place or better (approx 1:22.0) then I abort the race and try again.
Qualifying in 1:20.0 gets you in 2nd place on the grid, take 1st position by holding the inside line on the 2nd bend and don't lose your nerve or get too cocky!
Also, use the keyboard for better control (S,X,Left,Right).
Finally, I found a weird bug when you use the keyboard exclusively (including track selection), after a couple of aborted races, the next time you start, the qualifying lap gets aborted - seems an additional 'Escape' can be held in the keyboard buffer. I found using the joystick for track selection seems to overcome this. Maybe this is exclusive to the MSTE? Every version I tried exhibits the same problem, so I imagine it was a bug in the original game.