Ok so. Im 15 and i have a gf of 13y/o (for around 7months now, her mother found out somehow a month ago, we didnt want to tell her because as she says and i realised she is agressive and cannot mantain a good conversation without yelling.). Weird? i don't really know. I like her, i want to protect and be with her. My intentions are pure. I feel identified with her as i was when i had her age and didnt really felt secure of myself now i want to help her do it.

we grew close and now we're sorta in a relationship. Don't know if its love, but its the first time i have this sort of relationship with someone. Problem is, her mother doesnt accept us being together even though im a really good student, responsable and growing into maturity. Still. She is very very verbally aggresive towards my gf (13) so she cant talk to her without her yelling or bullying her.
what should i do to convince her that i only want the best for his daughter and how do i confront the fact that she is agressive and childish? i have both presenced those acts and my gf has told me, and that this is the cause of her being sad some times. Really breakes my heart and makes me wanna do something about it. What should i do?

Serious answers, killing her mother is not an option.