Asking Mom to Let You Shave Your Legs?

I'm 13 years old and I have started running in school for about 5 or 6 months now. I am obligated to wear shorts. At the beginning of the school year I was very uncomfortable with showing my hairy legs, but since the winter arrived I have been able to wear leggings for my athletics. I have addressed shaving with my mom many times, but she thinks that I am too young for shaving and that they aren't noticeable.

They are very hairy and dark because it is obvious and people look at my legs constantly. I also have issues with going to places with dresses or shorts because I can't have much fun with having to worry about hiding my legs. I constantly try my best to avoid clothes that reveal my legs and it becomes a problem in the summer when it's hot. Sometimes it's not even about the way they look, but how it feels. I usually feel like there are spiders crawling up my leg and I don't like the feeling. I am just really tired of her trying to avoid it and I was thinking about addressing it through a letter since every time I bring it up she shuns me out. Please help me, idk what to do in order for her to hear me out about my uncomfortableness.