I'm looking for a manga with a weak character put in a situation in which he/she has to start acting strong. And I don't mean the usual weak --> strong transformation or a dramatic change in the character's personality.

Some examples:
Liar Game - in order to survive in the game Nao had to stop being so gullible and dependant on others.
7 Seeds - Natsu had to do something about her shyness to stop being so useless.
King of Thorns - although Kasumi didn't change much, there were situations in which she had to do something, not just go 'kyaa!'.

I'm pretty sure Bokurano has quite a few scenarios like that.

+C: Sword and Cornett also shows that kind of 'forced change'.

Hatsukanezumi no Jikan - In a manner of speaking. But it's more of a 'clueless to more aware' kind of change than a 'weak to strong' change, so I'm doubtful about this.

Heptagon

Zippy Ziggy
Sakuran
Cat Street
Tenshi ja Nai!!
Majin Tantei Nougami Neuro
Nephilim (though he gets strong physically easily, mentally he has gradual change)
Please Save My Earth
Monster
Kiseijuu

Juuni Kokki o/ Twelve Kingdoms - The female protagonist must become emotionally stronger when she is left in an unstable and harsh world. Some of the things she had to overcome are her need for her peers' approval and her naiveté.

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Thanks All of these suggestions look very interesting, I have some of them on my wish list so I guess it's time to read them

I suppose the main character in Anatolia Story is going through something like that but it a quite slow process(2-3 volumes)