
You would think that being Superman's only surviving relative would be a sweet deal. Not So. Poor Supergirl-last survivor of Krypton's Argo City- is not greeted with the warm welcome you'd expect.

Let's back up a minute here. This all starts when Superman (using his Telescopic Vision) spots a rocket crashing to Earth outside of Metropolis. Upon arriving to investigate, he finds that the rocket was carrying a lone passenger: a girl in a dress that looks like his Superman costume. The girl explains that she is Kara, Daughter of Superman's uncle Zor-El. She was born in Argo City, a Kryptonian city that was hurled, intact, into space when Krypton exploded. Unfortunately, the chunk of Krypton Argo City sat on began to turn into the poisonous element Kryptonite.
To ensure his daughter's safety, Zor-El built a small rocketship and sent her to the primitive planet, Earth, which he had been observing through his "Super-Space Telescope". Having seen Superman and overheard radio broadcasts stating that he had come from Krypton, Zor-El decided the best course of action was to dress his daughter up like Superman...so he would know she came from Krypton...
So, once Supergirl explains this to Superman they figure out that they're actually first cousins. Does Supes take her in?? Of course not! That might somehow jeopardize his all-important secret identity. So he slaps a wig on her and sends her to an orphanage.

Yeah, 'cause no brunette girl could possibly be from outer space!
Superman insists that Kara refrain from being Supergirl publicly, asking her to wait in reserve as a "secret weapon". In an orphanage. Way to go, Superman. Good lookin' out there.
I have to say that Clark Kent adopting his orphaned cousin would not be a big clue that he was Superman. Even if he allowed Supergirl to go public (which, remember, he's not going to do) , Lois Lane is not that bright. I mean, if a pair of glasses keeps her from recognizing Clark Kent as the love of her life, a wig ought to make recognition downright impossible.
Also, Superman has a whole friggin' secret Kryptonian hideout at the North pole, the Fortress of Solitude (see Silver Age Man-Caves). Why can't Supergirl stay there? Why does she have to be all sad and lonely in an orphanage??
Despite his public image as a selfless hero, in his personal life, Superman looks out for Superman first.