Not "our" Supergirl, of course...this Super-Girl:
About 9 months before Kara Zor-El debuted, Superman #23 (1958) featured the arrival of a very similar yet different Girl Of Steel.
It's not hard to imagine that, given the similarities in costume and appearance, that this Super-Girl went over pretty well with readers, and inspired DC to create a permanent,albeit hyphen-less, version.
We begin with the boilerplate "oh, my life is too dangerous for me to ever take up with a mere mortal" spiel from Kal-El.
Well, Jimmy gets his chance, when he and Superman rescue a trapped archaeologist.
Quite a racket you have going there, Superman: "I'll rescue you if you give a priceless artifact to my pal as a 'souvenir.'"
And it really is priceless, because...
Well, this is the Silver Age DC Universe, so of course the legend is true!!
Thus:
Then the titans meet:
Look, I know it's sniggery and low-hanging fruit, but that panel just said that Jimmy wished a beautiful young woman into being by rubbing his magic totem last night. Wertham was right!!
Kal-El continues to be quite shy about his heterosexuality...
And it turns out that every time Super-Girl tries to help out, it backfires!
D'oh!
And...
D'oh!!
And...
D'oh!!
And then poor naive Super-Girl unveils Clark Kent's secret identity!
To see how Kal-El got out of this one, check out how he emotionally manipulates Lois!
Geez, no wonder he made Kara hide herself when she got to Earth--female super-heroes are nothing but trouble!!
Until, of course, Superman needs her, when some crooks throw some kryptonite at him while he's trying to save a train.
But...
"Must...use...all my remaining strength!"
Thus endeth the brief life of Super-Girl. We'll get a new one soon, but Superman will still treat her like crap...