Can you be badly injured in a car crash at 15 mph?

Yes. Certainly, the faster the speed, the higher the risk. However, critical factors include whether one was wearing a seat belt, the size of the car, the speed of the car that may have been involved in the crash, the size of the person, what part of the car was hit, etc.
Unlikely. You are not likely to be seriously injured if properly restrained with seatbelts. However, this does not account for another vehicle going at a higher speed. You should always drive safely and follow the driving rules.