I don't have much experience in non-BL mangas either. But if it helps, my non-BL collection:
For shounen+seinen
http://www.mangago.zone/home/mangalist/35626/
And my shoujo+josei
http://www.mangago.zone/home/mangalist/53203/
manga that I reaaaaally love so much other than BL:
Noragami. kami-kami
Haikyuu. sport - volley
Kuroko no Basket. sport - basket
Koe No Kitachi. shounen - romance
Donten ni warau. shoujo - family
HIRUNAKA NO RYUUSEI. shoujo - school life
Tonari no Kaibutsu-kun. shoujo - school life
STROBE EDGE. shoujo - school life
I honestly prefer josei or just generally sensible shoujo (most of the time, it's not). I recommend these bunnies:
http://www.mangago.zone/read-manga/cheese_in_the_trap/
http://www.mangago.zone/read-manga/totsuzen_desu_ga_ashita_kekkon_shimasu/
http://www.mangago.zone/read-manga/chihayafuru/
http://www.mangago.zone/read-manga/watashitachi_no_shiawase_na_jikan/
http://www.mangago.zone/read-manga/love_so_life/
http://www.mangago.zone/read-manga/otoyomegatari/
http://www.mangago.zone/read-manga/shishunki_bitter_change/
If you prefer more light-hearted comedy romance
http://www.mangago.zone/read-manga/mizutama_honey_boy/
http://www.mangago.zone/read-manga/afo_guard/
http://www.mangago.zone/read-manga/nijiiro_days_mizuno_minami/
My favorite shounen webtoons (I hate Naruto and stuff like those. These have more depths IMO)
http://www.mangago.zone/read-manga/kubera/
http://www.mangago.zone/read-manga/black_haze/
Hope that helps!
http://www.mangago.zone/read-manga/drug_candy/ - if you are into cheating spouses xD
http://www.mangago.zone/read-manga/lookism/ - first 2-3 chapters are not so interesting, but after he gets a new body it becomes really good
http://www.mangago.zone/read-manga/relife/ really cute and funny
http://www.mangago.zone/read-manga/gekkan_shoujo_nozaki_kun/ there is also anime
http://www.mangago.zone/read-manga/he_is_a_high_school_girl/
Shit I didn't read that you do not want tragedies and psychologies! Then definitely don't read Watashitachi No Shiawase na Jikan. It will tear your hearts out in just 8 chapters. And maybe kinda no to Cheese In The Trap too. It's mostly pretty funny but it can get dark at times (in a psychological way, tho nothing tragic or sad). But I really recommend CITT!!! It's like one of my all time favorite.
i'm not sure about my outlandish tastes so i don't rec as much as possible. but anyway, a reader of a lot of genres including yaoi, you can check
http://www.mangago.zone/read-manga/nodame_cantabile/
http://www.mangago.zone/read-manga/spicy_pink/ one of my fave cute yet josei manga
http://www.mangago.zone/read-manga/hachimitsu_to_clover/ (art students be forever poor)
http://www.mangago.zone/read-manga/noblesse/
http://www.mangago.zone/read-manga/the_god_of_high_school/
also there's a list i made that will probably help break you in straight mangas since it's got comedy, crossdressing and manly looking girls etc
http://www.mangago.zone/home/mangalist/45645/
Hello! I'm a true fujoshi and I've read looots of BL mangas. (And the truth is that BL is almost only genre that I've read..) I'm not saying that I'm tired of reading yaoi and shounen ai mangas but I just would like to read something else... :D
So, would you recommend some non-BL mangas? Shoujo? Josei? Shounen? Seinen? Fantasy? Supernatural?....
I'd like to read romance (and maybe fantasy/supernatural), but NOT harem, tragedy, psychological or anything really twisted.
Edit: It would be better if mangas were completed :)