ABOUT THE BUG:
When you're playing skyrim as usual, and you make a new save when riding your horse, after you reload this save, your horse started to fly all over the place, then congratulation, your character can't save the game while on horseback anymore, and sorry, it's permanent, there're no way to fix it. 

Since I can't find a lot of information about this bug online, I decided to share my knowledge about it, what cause this, and how to prevent it from happening. 

Long story short: This bug happened because you change your character's height mid playthrough, either intentional or not, this is the cause. 

Since skyrim doesn't actually has a height system, changing your character's height is basically changing your character's size, and somehow, the mounting physic is connected to the size of your character. When you start a new playthrough, you can use racemenu mod to change your character's height to what ever you want, and you won't get this bug and can still save on horseback normally. That's because the game save your height value when you first create your character, and will make sure you can save on horseback without issue. So the only way to get this issue, is by changing height mid playthrough, either by racemenu, or 'setscale' command, the result will be the same. So if you don't want to get this bug permanently, please, don't change your height during your playthrough.

But that's not all, I'm pretty sure most of the people who got this bug in their game, didn't actually change their character height mid playthrough, that's because there're two other way (maybe more) to get your character's height change by accident in the game.

The first one is a bug associate with ice wrath, here's how wiki describe it:

A lifeless ice wraith may spawn under the Dragonborn's feet. It takes place at Kagrenzel if the frost trolls are shouted down the hill with Unrelenting Force. A message will appear saying "was equipped" and "was unequipped." It is difficult to un-equip the ice wraith, after doing so, it will disappear, but its smoke will remain present. Something similar has also been observed to occur on PC: The head of an ice wraith will appear underneath the Dragonborn's feet along, with its signature frosty steam effect. It is most visible when the Dragonborn is walking on steep slopes or jumping, and it seems to have a habit of growing and shrinking repeatedly. If the Dragonborn is on a horse when this happens, they will be thrown high into the air and cease to be affected by gravity. It is possible to fly at high speeds across Skyrim with this glitch, as dismounting will remove the flying effect, but not the particle effect. A possible fix is to enter the following console command: player.addspell 000CAF8A and then player.removespell 000CAF8A.

For some reason this 'ice wrath' armor cause player's size to change, so unless you reload an early save, or try the method above, you'll get the flying horse when saving for the rest of your run. So watch out for this  'was equipped. ' message during your playthrough, whenever you get it, on a horse or on foot, don't keep playing like nothing happened. 

The second one is, if you change your character's height in racemenu when you started your new playthrough during your character creation, like I said, you won't get this bug later, with one exception! During the main quest 'diplomatic immunity', after your character take the ride to thalmor embassy, your height will be reset to default value, because you change your height when creating character, your character's height dose not match the default value of your race, reset the height means changing your height, which cause this bug. The reason for this is when using any furniture in the game, including the seat on the carriage, your height will be temporary reset to default value, but leaving the furniture will set your height back to whatever height you gave your character at the start of the game, so you won't get this bug by using carriage to fast travel. Sadly, when travel to the thalmor embassy, you actually get your height reset to default permanently (don't ask me why), so that's why after this quest you'll get this bug in your playthrough. To prevent this, you can use my plugin in download section, or do it yourself with SSEedit, by removing this keyword, your character will no longer reset height when travel with carriage, and of course won't get this bug when doing 'diplomatic immunity' quest. 

This is what you should do in SSEedit:



And of course, you can always save after dismount your horse if you like, it's not a very serious bug, just some inconvenience. If you currently already have this bug in your game, sorry, no matter what you do, how you change your height, how you tweak your game, you can't fix it. My guide is only for people who want to prevent this in their new playthrough.