The best advice I can give you is practice. Not very helpful, I know, but learn the course, learn the corners and how to take them. When to break, when to accelerate. And, maybe most importantly, keep your calm.
It took me a couple of hours of effective gameplay, but then finally I could get the bronze. That will be enough for now, anyway (I'm not very good at this game yet). By accepting that I couldn't win every time, I could see every race as practice, getting better and better results.
I can give you one other tip, though. The last "S" corners before returning to the oval course is a key to passing a lot of cars. Learn how to enter the first corner there and then don't be afraid to accelerate when you're coming out of it. If you do it right, you'll be able to go right "through" the last corner of the "S" and get out of it at high speed.
I'm sorry if my terms are a bit mixed up, English isn't my primary language and I don't know the correct terminology when it comes to racing.