If I hang out with a girl on the internet and we "hook up" does it count as a real relationship
Compared to one w/ a girl in "real life"??

No.

Define "hooking up with a girl on the internet". Be specific, please.

While a "real relationship" can be initiated by meeting on the internet, a relationship solely based on the internet is not a "real relationship."

By the term "real relationship" I mean a connecting of two souls by the process of communication, trust, love, mutual satisfaction (not only physical, but mental, spiritual, emotional), and mutual benefit.

"Hooking up", when used to simply refer to sex, is not a component of a "real relationship." Its simply hooking up.