In stud games 4th and 5th street refer to the 4th and 5th cards in the player's hand and not to the 4th betting round, since it is the 2nd or 3rd betting round (I am not quite sure exactly, because of the bring in). Check out the definition of a street here.
This term is used exclusively for stud games. In hold 'em type games I only ever heard about 1-3 streets of value. In hold 'em, streets are not named. Instead, we have flop, turn and river. The term used like this must have been wrongly carried over. Since the 4th street in stud games refers not to the 4th betting round, but to the 4th card, it can be concluded that those people use it in the same way for hold 'em type games.