Sometimes horse people will cringe over something nobody else notices. In the Game of Thrones series, Martin establishes that the rangers north of the wall ride garrons.
In the TV show they're riding...uh...I THINK those were Quarter Horses?
This, my friends, is a garron.
Specifically, this is a Garron Highland Pony, the larger type of Highland pony bred on the mainland, but the term used to refer to any larger (by pony standards), stocky, shaggy pony type - well suited to working in frozen and mountainous country such as you might expect to find north of the wall.
The descendant of the garron would be what we now call a cob - so the film makers should probably have gone and raided a Highland Pony or maybe Welsh Cob stud for mounts. Garrons would be used for riding, farming and general work in tough climates and rugged terrain - they weren't and aren't the most comfortable of mounts (although the Icelandic Horse, which is beautifully gaited, arguably qualifies) but they'll carry you all day where a higher quality horse would falter.