I go to an American private school they probably would call you down to the counselor if bruises were bad and frequent enough, but Gansey does say that he's made excuses for Adam when he didn't come to school because of his bruises, and there are plenty of acceptable excuses Adam could have made when he did come to school with bruises, considering two of his three jobs are very manual, and that he bikes to school(and there are probably assumptions that come with hanging out with Ronan & co.)
I mean, I guess that makes sense, but it still seems like they don't really give a shit. Like I'd find it pretty suspicious if some kid was always making excuses for why his friend/classmate was always coming in bruised, but I guess being that rich means he doesn't get asked questions.