Basic movement of player is not related to server TPS. You can move without problems even when server struggle to work. When server detects that you move to quick it just move you back to last position. And that's great.

Movement on horse is related to server TPS. Server need to refresh horse position every tick per second of server work. When server lags (because of large amount of mobs or other entities) it's really hard to ride a horse. Horse starts to slow down and randomly teleporting forward making whole horse riding experience.. just bad.

Why this is a bug? Well, bug is a mistake on programmers side that makes something not work like it supposed to, so that's why I trying to show you, that horse movement is made wrong.

Mojang could make movement of horse on client side and verify it on server side. The same way that normal player movement works. That would make riding horses a better experience on bigger servers with TPS issues.

Horses are a big step for Minecraft and Mojang. They still need some time to work on it and I really would like to see Minecraft horses on the best form it can be. This is a real issue and you need to consider rework the movement if you want it to be much bigger and better.
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TPS - Tick Per Second, amount of ticks that happens for one second of server work. Less TPS = More lag.