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In most major American debates, the candidates get a fixed amount of time to make their points. In theory, when the clock runs out they're finished. In practice, however, it's possible for a candidate to talk over everybody.
Why don't they just shut off the microphone when it's not someone's turn to speak?
The role of the moderator in a debate is to ensure that the debate flows smoothly and that it is conducted in a decent manner. Some of this role requires intervention (for example, a warning to stop speaking over an opponent wouldn't go amiss), but too much would interrupt the flow of the debate.
It's pretty similar in nature to refereeing/umpiring in some sports; often a referee will let some more minor infractions slide in order to allow the game to move more freely (especially thinking soccer here).
In respect to the suggestion of shutting off microphones, it's quite important that sometimes a candidate is allowed to make an immediate response, especially in the case of falsehoods.
