You BOTH are correct.
There are many canon versions of the Superman character. Publishers often change the storyline, which creates the conflicts. This Wikipedia article covers the publication history of original Action Comics, the series which introduced both characters. Your girlfriend is referencing the Silver Age storyline, while you're remember the Modern Age storyline.
Silver Age
Supergirl first appeared in Action Comics issue #252 which was published in 1959 (Silver Age).
Modern Age
The Brainiac Story Arc (Action Comics issue #869, published in 2008) has a different origin story. Brainiac stole the City of Kandor from Krypton before its destruction. It was later found by Superman on Brainiac's ship. Superman talked to Supergirl's parents.
Summary
As a whole, the DC universe is very big. I'm not certain I've covered everything. There may be other storylines where the ages of Superman and Supergirl are closer. Without specifying the particular storyline, the debate is meaningless.
