Minecraft 1.14.4 Horse Not Going Through Nether Portal
This seems to be a bug. I was able to reproduce it, but it only happens sometimes to me. I created a report here: https://bugs.mojang.com/browse/MC-158146
As a workaround, try pushing the horse into the portal instead of dismounting inside the portal.
I had the same issue and it appears it was caused by blocks blocking one side of the portal on the other side. If you make sure to clear enough space around the portal, it should go through first try.