Grow

Grow is used before adjectives to talk about slow and gradual changes. It is more formal than go or get.

Before you notice it you will grow old.
The night grew dark.

Grow + infinitive can be used to talk about gradual changes in attitude.

She hoped that someday he would grow to love her.

Turn

Turn is used to talk about visible changes. It is common before adjectives of color. Examples are: turn blue, turn red, turn green etc.

Leaves turn brown in autumn.
She turned red and walked out of the room.

Turn into
Turn into can be used to talk about a dramatic change in the nature of somebody or something.

Everything that King Midas touched turned into gold.
He is a nice guy, but when he gets angry he turns into a monster.

Turn is also used with a noun or adjective to talk about changes in occupation, religion, politics etc. Note that no preposition or article is used before the noun or adjective.

Towards the end of her life she turned Catholic.
He worked in a bank for twenty years before turning actor.

Fall

Fall means ‘become’ in a few fixed expressions. Examples are: fall ill, fall asleep, fall in love etc.

She fell in love with an old man.
The baby has fallen asleep.