Do you have an external flash, a hot-shoe mount? How about a two step ladder? The reason I ask is because either of them would be of a benefit and having both is a bonanza for this type of shot.
The sky is quite bright with clouds, but does have blue streaks running through it. Add to that a very dark subject and the dynamic range is vast and likely beyond your camera's capabilities. Don't know, just a guess. Having the external flash will allow you to under expose the background a couple of stops giving much more detail in the clouds and expose the horse properly, giving much more detail in the shadows.
The two step ladder elevates you obviously, but this is important because you want to be eye level or slightly higher to the horse. Barring a ladder out in the paddock, is there a wooden fence that you could use? The idea is to reduce the amount of featureless sky by having a higher PoV (point of view) and thereby having less of it in the frame.
If neither of these options are available to you, having the sun to your back will probably be your best bet for a good exposure.
Lastly, try to find a background that is thick with a tree, a bush or a shrub (anyone remember Lady Bird? and I ain't talking about Hank Hill's dog). In your second photo, the tree just under the horses nose is a fine example of what I mean. If you can something like that which covers the entire background and by opening up your aperture, you will certainly isolate the horse from the background. Read make the subject pop.
Just my 2¢.