If you are socially anxious, I will always suggest doing cognitive therapy. Without changing our thoughts and feelings we will not be able to change how we react to situations - which would be anxiously if you indeed socially anxious. People may suggest just pushing through it, forcing yourself, but that idea does not apply to seriously socially anxious people who force themselves every day and still feel terrible. So, step one - cognitive therapy to change how we think about situations, about life really.
Using rational therapy strategies to question your ANTs here, I would say that actually I see many young people at my last gym. It was a YMCA and there were always groups of younger people from elementary school even. Some were organized in classes, but some were by themselves in the gym. Perhaps your gym does not have a younger clientele, but personally I do not think anything strange when I see a young person at the gym. Seems like all ages go to the gym nowadays. And if you are nervous about being younger, you might call or email the gym about rules for age groups - just to help you feel better rationally.
I know what you mean about the gym environment making you feel nervous. People can and do look at other people. I also find that most people at the gym are really focused on their own thing, their workout and not yours. Part of gym etiquette is to sort of allow people their private space to exercise without too much scrutiny. That's how I behave and think at the gym. I'm focused on me and not other people. I'm friendly and polite at the gym, but I'm there to exercise and not really to talk or look at others.
I'm just trying to give you some ideas you might use to help you think differently about the situation. If you must go to the gym, if you feel that is the only option and that you cannot discuss good reasons to not go to the gym with your mother, then facing this situation rationally and with a healthier mindset to use the experience to grow and use healthy thinking strategies is hopefully a way you might approach this.
It just helps me in all situations to apply the cognitive therapy.