A bit of shy/a bit shy.

[Rollercoaster1]

Are these both phrases correct?

 

[emsr2d2]

The first is not grammatical.

He/she is a bit shy.

"Shy" is an adjective, not a noun, so you can't have "a bit of shy".

 

[Rover_KE]

Are [STRIKE]these both[/STRIKE] both these phrases correct?

`

 

[Rollercoaster1]

`

If that's the case of correcting my phrase, I would be happier if you had corrected it as "are both of these phrases correct"?

 

[engee30]

If that's the case of correcting my phrase, I would be happier if you had corrected it as "Are both of these phrases correct"?

Well, there's gratitude for you. :-|
Your correction does not make much of a difference there, to be frank. Americans appear to use that version more than the Brits do, though.

 

[Tdol]

Regardless of variant, the original did need correcting.