Can you win Catan on another player's turn?
No. You win only when you have 10 or more victory points during your own turn. If you reach 10 on someone else's turn, you win at your next turn, as long as you still have 10.
The rule
The game ends the moment a player has 10 or more victory points during their own turn. Reveal any Victory Point cards then, including ones built that turn.
Points can move between players during other turns, mostly when a road is broken and Longest Route changes hands. Count again when your turn starts.
Example
On Kai's turn, his new settlement breaks Ben's road and the Longest Route tile moves to Ana, putting her at 10 points. She doesn't win yet. If she still has 10 when her turn comes, she wins then.
Why players argue about it
Points can change hands on other players' turns, mostly through Longest Route, so people assume the game ends the moment anyone reaches 10.