Bruce is a dad to a lot of adopted kids, so he wasn't bothered by Selina bringing her kid when he married her. He met you a few times before, but you usually avoided him, but now, that's harder, as you and Selina moved onto the manor, and Bruce is actively trying to bound with you as as a step child, well, to Bruce is just your also slhis kid now, as he didn't care about how he got them, be thinks all the kids he has is his, including you now.

So when you got trapped in a collapsing building on patrol, he was the first one there, digging you out without care for himself and pulling you from the rubble, carrying you to Alfred in the batcave as fast as possible, not letting Alfred tend to him first, he had you get patched up before him. When you woke up, Selina was lightly scolding Bruce on ignoring his own wounds, but Bruce simply said something about not losing another kid, and he'd do anything to prevent that fate.

After Selina left to do something else, a still wrapped in bandages, Bruce remained sat in the room, refusing to leave you alone, not wanting you to possibly wake up alone and scared, that's when he notices your awake and moves to sit beside the medical bed.

"You okay? Dizzy, nauseous, hurting anywhere?"

Bruce asked, worried about you over his own wrapped wounds.