(Bat Dad on this one because I like that headcanon about him)

You would think being related to Dr Hugo Strange is not much of a bad thing but it is, the emotional and physical abuse, being neglected all the time, needing to be perfect in public and treated like one of his patients. It does a lot of things to a person, even drive them to their breaking point. The villains of Gotham may have pity for you and try their best by letting you stay with them for awhile, giving you old clothes of their and giving you basic life skills. There only a bear minimum of comfort they can give you, but it doesn't erase the trauma and fills in the void of what you need.

Batman has seen this all, how Dr Hugo Strange acts like a loving family member to you around people in public but the moment you two are alone. You are getting yelled and every mistake you made that day would be pointed out, how you weren't even good enough to even be related to Hugo Strange at all. Even when Bruce Wayne isn't Batman at the moment, he sees how you are treated and how everything is an act for the public eyes.

Offen when Bruce Wayne is out of his Batman personality and sees you in a public setting, he tries to show you what a real parent figure looks like. (He pretty much ignores Hugo Strange when he standing right there, knowing full well all the sweet talk is fake). As Batman he tries to keep you out of trouble and keeps you on a good path, he knows the villains aren't good role models but he doesn't really trust them. If he could (and would probably do it anyway) he would adopt you as his own child.