After accidentally killing 2 of my 3 horses from drowning I began to brighten up. Why are they drowning? Usually mobs will jump up in the water and that helps protect them. Do horses drown because they are weighed down by the saddle?
They are weighted by the rider. It's hard to say why they don't always swim up - they just don't, probably a bug, but maybe it was just written that way - who knows?
Anyway, don't go swimming on the horseback. Have a lead with you, and as you approach a body of water, dismount, leash your horse and swim across pulling it after you.
Edit: It's a bug.