I can't believe I didn't think of something this simple earlier. Horses absolutely can swim - and some of them even seem to enjoy it. They can even swim while carrying a rider, although this is a lot of effort for them and not something you want to do for a long distance. (Also, if you can't swim yourself, don't take a horse swimming. You can't use a saddle as it would get ruined and it's very easy to slide off). Of course, in an emergency, your characters may want to swim horses across a river with tack still on. But it will really mess up leather tack.
Some horses can and will swim for the sheer fun of if there's a pond deep enough in their pasture. And the Chincoteague pony roundup is called the "pony swim" because the ponies are herded across a seawater channel between two islands that is deep enough for them to have to swim.
Image source: U.S. Coastguard via Wikimedia Commons.
Before a bridge strong enough to carry mules was built across the Grand Canyon they used to dismount and swim the mules across the river from the south side to Phantom Ranch.