After the Crisis Karen felt devastated, believing she wasn't Superman's cousing but a fraud. However, Kal talks with her and makes known that she'll always be his cousin as far as he's concerned. And then they hug. Aww.
In Supergirl issue 20, Karen and her counterpart talk:
SG: I'm sorry, Kara. I didn't mean anything by it. It's just that this is the one place where I shouldn't have to worry about anything. So this is all incredibly frustrating. And when I get frustrated I say some pretty stupid things. PG: I remember when I was like that, too. SG: That's good to hear. If I can grow into at least half the woman you are, then maybe there's some hope for me...