You BOTH are correct.

Actually, [there are many canon versions of Superman character exist][1]. Publishers often changes storyline (which are conflicting to each other). See [this Wikipedia article][2] to get a look at publication history of original *Action Comics* (it is this series which has introduced both characters to the world). Your girlfriend is referencing *Silver Age* storyline & you have seen such things in *Modern Age* storyline.

Supergirl was first appeared in the world by *Action Comics* issue #252 which was published in 1959 (Silver Age). According to it:  
![enter image description here][3]  
![enter image description here][4]  
![enter image description here][5]  
![enter image description here][6]  
![enter image description here][7]  
![enter image description here][8]  
![enter image description here][9]  
![enter image description here][10]

While covering `Brainiac Story Arc`, I once found a different storyline of Supergirl in *Action Comics* issue #869 which was published in 2008. Brainiac once stole the City of Kandor from Krypton (before its destruction) which was lately found by Superman on Brainiac's ship. Superman talked to Supergirl's parents..  
![enter image description here][11]  
![enter image description here][12]  
![enter image description here][13]  
![enter image description here][14]

As the whole DC universe is very big, its impossible for me to cover everything. There's possibility that there may exist another storyline around Supergirl saying both have same age. So, such type of debates are meaningless.


  [1]: http://scifi.stackexchange.com/questions/22733/as-of-2012-how-many-different-canon-versions-of-the-superman-character-exist
  [2]: https://en.wikipedia.org/wiki/Action_Comics
  [3]: https://i.sstatic.net/tK1WK.png
  [4]: https://i.sstatic.net/JDCga.png
  [5]: https://i.sstatic.net/5yTcd.jpg
  [6]: https://i.sstatic.net/WtIxV.png
  [7]: https://i.sstatic.net/5QCHZ.png
  [8]: https://i.sstatic.net/EbodW.png
  [9]: https://i.sstatic.net/GmVg4.png
  [10]: https://i.sstatic.net/fSvsv.png
  [11]: https://i.sstatic.net/5VNnj.jpg
  [12]: https://i.sstatic.net/OGiz7.jpg
  [13]: https://i.sstatic.net/BfFBp.jpg
  [14]: https://i.sstatic.net/vT2gb.jpg