To exit, you must first swim down and pull on each of the 4 blue lights to activate them and drain the water from the room. This sequence is timed, so you have to work quickly; otherwise the lights reset and you have to start over.
Repeat the same procedure for the third and fourth lights. You may have time to come up for a quick breath, but it's possible to make it without surfacing. For me, swimming up then down again was more difficult than just staying level with the lights.