You can't "make" a child not shy. It is intrinsic in them. They usually outgrow it.
My daughter was like that. She is also very bright, social, and a mature girl... at home. In class at preschool, she didn't "reveal" these things, because she was shy. Its okay. Her Teachers fully knew her capability.
She is now in 2nd Grade... but since Preschool and until now, each Teacher has said how wonderful she is BUT she is "shy." I answer by saying "I don't see that as a problem...." and she is a great kid and student... and LOTS of great leaders & scientists & Geniuses in our world were shy as kids. Nuf' said.
Next, just teach her to be proud of herself, to be herself always, and to know that everyone is different. Do not lead her to believe that something is "wrong" with her... that will only make a child turn inward. More.
We simply taught my girl that its okay... because she really KNEW herself from such a young age...and we taught her that she is fine... that when she does feel comfortable... she CAN play with others or not. No biggie.
The Teachers did not "force" her socially. But they would comment on it.
My girl is now 7, and she has outgrown a lot of it. But she is still a tad shy. BUT, the silver lining it this: She is a confident little girl who knows herself WELL... and can assess other kids well and know who is nice or not... AND she is not a "follower" and can wisely choose friends, consciously. I am proud of her. You see... all that "shyness"...actually was beneficial because she was a keen observer... and she formed her OWN ideas and knew values that way... and how to be herself... her own person. Which is what we value in our kids.
So, use this as an "opportunity" to teach her about people, how to "discern" people, how to be herself and not feel bad about it, how NOT to "compare" herself to others, give her confidence about her shyness & that it is not a "negative" thing... and don't label her.
That is what we did with our girl.
She is actually very well adjusted... and one girl even told her "you're real popular...." and they want to play with her. But the thing is, my girl is not influenced by those things... she values people and even befriended an Autistic boy that was mainstreamed in her class. She has great "empathy." She is just herself... not a copy cat of the other kids. She "chooses" her friends wisely... which is what as a parent, I am most proud of her for.
ALSO teach her, or instill within her... that if other kids do not play with her... do not take it personally... just go and play by yourself or choose another play mate. No biggie. My daughter, NEVER took it "personally" if other kids did not play with her.... because she knew herself. She simply did something else or made other friends.
My daughter, takes Karate. She LOVES it. BUT, we did not make her or force her to take lessons. She asked us to, all on her own. It is her thing. And, we would have never suspected that she'd want to take Karate... but it provides a value to her... and she finds it fun.
All the best,
Susan