Looking at the map again, that would make sense. We'd not observed the shape of the water closely before writing that last OoC post.
We now imagine that boats would be on the way, but not deployed enough to form a proper cordon, if that is even possible on water without full-sized ships. Across the water will be the most viable escape route. The Kalesian empire might have sent a pretty large army, but it cannot effectively surround the entire lake/ocean in time to stop smaller groups escaping. 

We would agree that there are probably numerous rivers not shown on the map. Only the largest would be visible on such a map. Logic dictates that there is some sort of flow westward out of that water. It is either that or the region there is hot enough to evaporate the excess water, which would be rather implausible.