The Lord of the Rings is a VERY low-magic world. There's no wizards blasting each other with spells whatsoever, magic is subtle or barely present at all, yet it's THE defining work of High Fantasy literature, and still the work by virtually all other High Fantasy literature is compared.
Conan the Barbarian by contrast has FAR more overt magic. Sorcerers, wizards and magic are VERY prominent (ever other villain is a sorcerer of some sort). And once again, it's pretty codified Low Fantasy.
High vs. Low fantasy has NOTHING to do with quantity of magic, but by the scale of its themes, setting and plot. This plants GoT FIRMLY in High Fantasy.