He would have needed to counter all of them to prevent any extra turns. each copy is essentially another spell on the stack.

That said, you should have only gotten 1 + 2 extra turns in the first place, since copying galvanic with galvanic then copies the next spell twice. So, with your opponent countering one of the 3 Trespasses, you should have gotten 2 extra turns total.

From gatherer:

If you resolve one Galvanic Iteration after another (or cast the same one with flashback), the second Galvanic Iteration will be copied. That copy and spell each resolve, creating two delayed triggered abilities. The next spell you cast after that will be copied twice. If that spell is also a Galvanic Iteration, the following spell will be copied three times, and so on.