The topic says it all. Someone who sees ghosts or knows the future or something like that--but it has to happen in a normal "real" world like in Ghost Hunt. No huge organizations battling monsters, etc. Like, there is our normal life, and then this mysterious character in the middle of it. Or they can be a fake psychic, like in that dorama where a fake TV psychic investigates non-supernatural murder.

Pretty please? ^__^

Mekakushi no Kuni
いっしょにねよ うよ
I'm sure they are lot of others stories that fit the requirements but those are the only one I've read.

Shinrei Tantei Yakumo

Haruka and Yakumo have a kind of dual protagonist thing going on, focuses more on Haruka though.

________________
Is this me in a nutshell?
Pretty much.

Read webtoons??
Supernatural investigations department - S.I.D
Sorry dont know how to do the clicky thingy

Thanks! More recs are welcome!

Tasogare Otome x Amnesia
Anata ni Hana o Sasagemashou
Yamamoto Zenjirou to Moushimasu
Tsuiteru Kanojo (ENJOUJI Maki)

________________
~4400 manga completed
~1700 ongoing...

Not all of these are in normal settings, but they're really good and they're not about huge battling organizations:
Hyakki Yakoushou
Natsume Yuujinchou
Mushishi
Helen ESP

This is the reverse of what you're asking, but quite interesting:
Iris Zero

A stretch (fits the 'normal setting but you have an unusual character' theme):
Alyosha! - Actually an assassin trying to live like a normal girl. Pretty funny.
Warau Kanoko-sama - Nothing supernatural, but the female protagonist has a hobby of observing everyone and occasionally blackmailing them over it.

None fit your request exactly, but maybe you'll enjoy some of them. ^^