Are you looking for something fundamentally D&D-like, in the vein of a low-magic OSR heartbreaker?
Or do you want to go further afield into more narrative style games, a la Fate and BitD? They're very different types of games which don't always appeal to those with deep D&D backgrounds.
Despite the ubiquity with which people ask, finding a D&D style game with low magic is surprisingly difficult. I think the simplest reason is that D&D is fundamentally a game about advancement; without magic to explain how the characters progress, there's a pretty big hole for the system to fill and imperfect rationales as to how to do it.Definitely not FATE.
Conventional fantasy with a much smaller emphasis on spellcasting and magic in general.
Adventures in Middle Earth does rare magic high fantasy really well (unsurprising since it is built to emulate Tolkien) on a 5e chassis so it is a relatively easy transition for D&D players. You'll have to do a little work to separate the Tolkien parts but that can be accomplished mostly by reskinning things and adding what you want rather than tearing out whole pieces.Definitely not FATE.
Conventional fantasy with a much smaller emphasis on spellcasting and magic in general.
Adventures in Middle Earth does rare magic high fantasy really well (unsurprising since it is built to emulate Tolkien) on a 5e chassis so it is a relatively easy transition for D&D players. You'll have to do a little work to separate the Tolkien parts but that can be accomplished mostly by reskinning things and adding what you want rather than tearing out whole pieces.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.