I truly believe I had a guardian angel growing up. My brother and I rode bareback into our pond often with our ponies when we were 9 and 11 years old. We knew to give them their heaad and never interfered with where they wanted go. The pond was probaby was 12 feet deep or so. They took very good care of us and they and we enjoyed this often. I would not have done this on most horses, but these were careful with us.
Swimming with your horse or pony is much easier if you go across slow moving streams with narrow central deep sections. Turn your horse with it's head facing down-stream in the deeper center of the water. The current will help your horse move through the water with little effort. If the horse panics, turn the horse to the banks of the river and exit. I've trained my equine buddies to carry me bare back and respond to neck reining while wearing a rope halter. My Arab and my Paint allowed me to grab their long tails so they could tow me behind them.