Sadly, I had the same experiences as Angelique in that sense, basically didn't think I was mature enough, or didn't know enough. Granted, our experience is anecdotal, but it is quite a common attitude. Further, some won't see you until 18, for insurance reasons, sadly know that first hand too well.
I've been in your shoes, and my mother gave me the same exact saying at one point, Maka. What did I do? I made the fact that I was/am/will be female now, and I will be forever, with the disaster that was becoming me, physically, emotionally, and in stability. At every opportunity. Obnoxious? Yes. Got the point home? Also yes. She now 100% supports me. While I can't say that will happen with you, sadly, your mother wants to see you be happy, make the right decision. She's worried for you. And if you show that your concerns are rooted in having T surge through you, then just maybe she will see your point.
In fact, you should talk to your mother about her concerns. If you see it from her vantage point, she doesn't want you hurt, wants you healthy. And if you talk with her about her concerns as equal, valid, and worthy of discussion, that, too, is an absolute maturity sign. And you should do it, you have nothing to lose by doing so.
The other idea, outside of your personal push, struggle to be presented, is to show her documentaries, articles, etc. of those around your age (I also did this) starting hormones, to show that it entirely is possible to do so, and that not only is it possible, but how happy, how vibrant these lives seem as a result.
PS: If you're around where I am (New England, USA), I can give you a fair share of resources of doctors who treat specifically TS youth. While it may not be the actual estrogen, given you aren't at age of majority yet, lupron/blockers are an alternative countless times better than the T.
If you have any questions about transitioning relatively young, the medical system under 18 (at least, in the USA), etc. shoot me a PM. I wish you the best with your mother, and while it may seem hard now, the rewards are so, so worth it.