A river is a ribbon-like body of water that flows downhill from the force of gravity. A river can be wide and deep, or shallow enough for a person to wade across. A flowing body of water that is smaller than a river is called a stream, creek, or brook.
As this became colloquial, 4th street became known as the turn, while the river is a much older term that comes from before community card poker games were even played. Sometimes cheaters would deal themselves and their partners an extra card after the hand was supposed to be complete.