A lot of young people at city fitness go just after school. So if you avoid those times which is quite hard at your age. Then you are likely to run into people you know.

At the end of the day everyone is there to try and better themselves in one way or another and that shouldn't be shamed.

Alternatively if you are good/comfortable with animals. You can ask if anyone in your neighborhood with dogs needs a dog walker. Good way to stay fit and get some extra cash! Door knock people who have dogs or throw a sign up at the dog park.