I think at 13 it could be pretty difficult to lose weight healthfully without your mom's support. At 14 I chose the UN-healthy option of starvation but I can tell you that while I did lose weight, I looked like crud and my hair started falling out. <- that was a bad choice!
It is possible (well, very likely) that your mom feels guilty about your weight, and so, is being defensive. Maybe she would be more helpful if you looked around on here and figured out what weight loss plan you think you could live with long term and then had a talk with her about it. I think approaching it in a way like "I know you love me and only want the best for me, so I knew I could come to you....." would really help get her to take you seriously and not feel defensive. Try not to place blame but tell her how you are feeling.
If all else fails- keep coming here!