we say "motor racing" in the uk - no one would ever say "car race"
The purpose of learning a language is to accurately communicate with other people. Assuming that other people think the same as oneself frquently leads to miscommunication. For example, in talking to one group of people a "race" could be referring to a "horse race" while to another person or group in another geographic location it could be a "stock car race", a "drag race" (auto - not drag queens), or a bicycle race, etc. The point is --avoid being misunderstood by being specific about what you are attempting to communicate.