When I ride my horse and my donkey I get thrown off after just a few steps (maybe 4 or 5 steps). However, I can ride a pig with no problem.
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Yes, both horse and donkey are tamed. Didn't have any issues with being thrown before Phase 1 Aquatic Update.
Auldrick
This is probably a duplicate of MCPE-33152, which caused the game to "forget" who was a tamed mob's owner. If so, the problem only applies to animals you tamed prior to 1.4.0. It was fixed in 1.4.2, but the fix only applied to newly tamed animals. Mojang was said to be looking for a way to restore the lost ownership.
To test whether this is the same issue, you can tame another horse and see if it will let you ride it, or tame a wolf, ocelot, or parrot and see if it will sit for you. If so, it's the same issue. Otherwise please comment about what you tried and what happened so that we can look at this in more detail. (The ticket will automatically reopen when you reply.)
Have you tamed the horse? if not that's your problem.