Normal Humans with Special Powers, you say? Well, I can suggest you animes like: "Afro Samurai" and the movie "Afro Samurai: Resurrection"; "Ao no Exorcist"; "Monogatari series (Bakemonogatari, Nisemonogatari, Nekomonogatari, Monogatari 2nd season, Kizumonogatari)"; "Deadman Wonderland"; "Inou-Battle wa Nichijou-kei no Naka de"; "Kami Nomi zo Shiru Sekai" - 2nd season "Kami Nomi zo Shiru Sekai II and 3rd season "Kami Nomi zo Shiru Sekai: Megami-hen"; "Kokoro Connect"; "Kore wa Zombie Desu ka?" and 2nd season "Kore wa Zombie Desu ka? of the Dead"; "Kotoura-san"; "Kyoukai no Kanata"; "Mahou Sensou"; "Noragami! and 2nd season "Noragami! Aragoto"; "Nyan Koi!"; "Suzumiya Haruhi no Yuuutsu" - 2nd season "Suzumiya Haruhi no Yuuutsu (2009)" and the movie "Suzumiya Haruhi no Shoushitsu"; "To Aru Majutsu no Index" - 2nd season "To Aru Majutsu no Index II" - the movie "To Aru Majutsu no Index: Endymion no Kiseki" - spin-off 1st season "To Aru Kagaku no Railgun" and 2nd season of spin-off "To Aru Kagaku no Railgun S"; "Yamada-kun to 7-nin no Majo" -
Those are animes I watched that have at least 1 character who has magic in his hands without being necessarily being a "God". They are different kinds of animes and may not be to your liking but please take a look!