at which point you lose troops when the Westeros card checks for supply?
When the Supply card appears, the players should adjust the supply track according the number of barrels you have in the controlled zones.
The track is just updated when the supply card of Westeros appears or another event.
In example: You start in 2nd position in the supply track (unless you are a Stark), while you advance in the game, claiming new zones with barrels, when in the westeros phase appears the Supply card, all the players should count the barrels that each one controls and advance or back from the supply track
Edit: You cannot exceed your number of units indicated in your supply track, If you lose zones with supplies and the westeros indicate you should update the track, you should remove units until you have a legal size of armies