How did the young man take his turn? (spoilers)
Sure it was the old man's turn... except he was incapacitated at the time, so he couldn't do anything, which is why the bartender allowed the young man to continue the game. In a normal game of billiards, if a player is knocked out and no one is able to substitute for them, then the game would naturally end. In Death Billiards, however, ending the game was not an option, so the young man decided to take over the old man's turn and finish the game