IIRC that version of Supergirl landed on Gotham and Bruce was the first person that encountered her and after that he felt responsible for her. >>114377869
is out of context but she was trying to sell him Kryptonite tech to get money and didn't want to get it for free and being strictly business but he sees through her that she needs money.
This was also during that weird arc where she thoughy she was sent to kill Clark so she was avoiding him and Bruce and Diana were the other peoole she was closest to so she bonded note with them. Kara was also an unofficial member of the Batfamily. Bruce gives her some fighting training, she ocassionally hanged in the Batcave, and Bruce, Dick, Tim, Steph, and Alfred knew her and Damian was tsundere for her.
There's a comic where she recieves a present in her house and she notices she can't see through it because it was lined with lead. She opens it and sees a paper that only says "Gotcha" on it. Later Bruce calls her and berates her for opening the present and tells her to be smarter and that it could've been from an enemy and had kryptonite. She gets annoyed at him for being weird.