I watched a show 6-8 years ago about people that got into robots to fight kaiju-like creatures. It was kind of like Pacific Rim. The main character was a guy who was school. A girl that tried to control one of the robots had gotten severely hurt when fighting a kaiju creature.
This sounds a lot like Neon Genesis Evangelion. It aired on Adult Swim in 2005 and has popped up on their schedules repeatedly since then.
The main character, Shinji is a
mopey little sod schoolboy who's forced to pilot a gigantic mecha and fight kaiju-like "angels". One of the other protagonists is a young girl called Rei. She gets seriously injured about once every other episode.
In Dual! Parallel Trouble Adventure our main character sees something that no-one else sees, giant robots fighting other kaiju-like things. He meet girls (which is interesting with his story) later he move to parallel universe where he takes over a mecha because the girl who pilot him was injured
Fafner is almost exactly kids-fight-with-help-of-mecha-vs-alien-monsters.
RahXephon again sees a boy see some monster attack city, he meets a girl who leads him to strange undergrounds with a big robot which he use to fight monsters
Adult Swim also showed Big O. The girl was an android named Dorothy. The main character was officiakly a "Negotiator", but was more of a hardcore detective and could handle himself in a fight, even when not in Big O. One unusual element was that the pilot of the mech would be judged by the mech. Each episode had the town attacked by some form of kaiju.