Background info- I'm 14, my little sister is 4, my older sister is 15, my sisters don't usually get along with each other because my older sister doesn't like kids and has a short temper with them. I taught both of them most things- my little sister how to walk and some social stuff (basic social stuff, I'm not traumatizing my baby sister), my older sister much more since I was with her long, I taught her how to control her anger, how to be nice when she needs to, and more I won't mention. My parents are present, just not fit to properly care for kids so I decided to do it myself.

Now the situation, me, my mother, and my sisters went on a car drive to the beach (I admit I was sort of on edge because I hadn't slept more than 3 hours in two days), I was on the passenger seat, my mom was obviously the one driving, and my sisters were in the back (dumb idea, I'm aware). They both started arguing over my older sister moving a bag that was in between them (some context there is gone because it isn't relevant to the story) but I didn't intervene for about 20-30 minutes, mostly because my mom was trying to do it.

My moms attempts included a bunch of telling my older sister to stop, comforting my little sister, and going right back to berating my older sister, which wasn't the tiniest bit effective as my little sister was still crying and my older sister was still being mean to my little sister- so I finally stepped in, tapping my mom on the shoulder for her to let me do it (this happens every once in a while when my sisters have a fight and father isn't there to be horrible enough to stop it) and she nodded.

I asked my older sister to let me speak to my little sister for a moment, and promised I wasn't blaming her and would hear her out too. Then I turned to my little sister and calmed her down slowly before asking why she was crying, in the end I came to an in-between after more or less 5 minutes and told both of them to call me if something happens again before fighting so I could fix it, plus gave some reasoning. My older sister said something again so I calmed my little sister down and repeated that she should call me first things first if our older sister said anything, she agreed and calmed down again, but my mom gave me a look and said she was their mother and that I was undermining her (quiet enough so that my sisters didn't hear) and I responded something along the lines of "i fixed it, didn't I?". There was an uncomfortable silence for 10 minutes after and I'm having serious doubts over whether it was the right thing to do and say or not, since my mom already struggles with thinking she isn't a good mother.

I'd appreciate all help and opinions, if you think I'm the AH please tell me why so i can be better next time, I genuinely came here for a chance to get unbiased criticism aswell as know if I was right or wrong.