You may want to read the books rather than just watching the films.Um ... you might want to check your sources. Conan particularly (at least as written by REH) fought mostly beasts and men, with the occasional otherworldly horror. He's the epitome of the low-fantasy model that you've just decried.
While Conan fights a lot of men there are also great hordes of demons, giant serpents, devolved humans and other monstrasities.
About the only thing he fights which might be called a "normal" animal is the occasional giant ape thing.