You may misunderstand when the condition is checked, or what the difference to Outpost is. The condition is checked when Voyage is played and resolved, not when you are in or about to enter an extra turn. Outpost has another clause in its text that makes it work differently from Voyage.
What Voyage does "prevent" is that a player plays Voyage, then takes their Voyage turn and during it plays another Voyage. Of course they can do that, but the second Voyage checks that the previous turn was also that player's, and fails to give them a second extra turn.
What Voyage does not prevent is a player playing multiple Voyages during the same normal turn, and them then getting as many extra Voyage turns afterwards. Because each time a Voyage is played during that normal turn, the previous turn was not from that player, so the extra turn is given.
That's exactly what happened in your case. On their normal turn 14 your opponent played 3 Voyages, so they got 3 extra turns afterwards, which is perfectly OK with the rules.
Some players are confused about this because they know Outpost, as Seaside card which is the only other playable card that grants the player an extra turn (other effects are from landscape cards like Events, Projects or Allies). Outpost does indeed prevent both scenarios above from "working", including the second one of playing multiple in a normal turn to get multiple extra turns.
But that is because Outpost uses the phrase "If this is the first time you played an Outpost this turn,..." This is what prevents multiple Outposts in one turn from "working", but that phrase is not on Voyage!