Well, first off, you don't have that handy-dandy bar along the top of the screen that tells you when you're running out of breath like you did in the other games. I was saddened by this myself, as the first time I went for a little swim thinking "oh cool, I have water breathing!"...until I suddenly found myself belly-up in the middle of the lake, having my body picked at by seagulls. Whoops. If all else fails, just swim up, and do it on a moderately frequent basis (unless you're an Argonian, which I'm assuming you're not since you drowned).
Now, when you say you can't see underwater, what do you mean? Let's face it, the water in Skyrim is a touch on the murky side (mmmm, algae), it is rather difficult to see in it, but I always found that to add to the realism. I guess I'm just not understanding what being able to see has to do with the breathing...