Lois raised an eyebrow. That was a dramatic overstatement, but yeah, she got it. "He doesn't trust many people; maybe anyone, I don't know. But how is he going to trust you if you don't give him the chance?"
Granted, getting Batman to trust her probably wasn't high on Kara's 'To Do' list, but the man was smart.
"Hey, you don't have to convince me. There's a lot about him I don't know, and I don't want to. But I do know that he never does anything without a reason. He could have done nothing and let you live with us, which would have enabled him to ask Clark to keep an eye on you 'just in case'. Instead, he gave you a place of your own, away from Clark's hovering, where you can have a little freedom.