Is there a word to describe female between 'girl' and 'woman'?

Solution 1:

There's "ingenue" which has the connotation of innocence. If you want to go less formal, there's "chick" or the latin-flavored "chica".

Finally, as a female, I can attest that it is always a badge of honor to be called a woman. :-)

Solution 2:

The term "woman" refers to a female who has reached adulthood. The term "girl" generally refers to one who has not.

That said, the word "girl" can be used loosely to refer to any female human being:

Granny was a spry old girl, well into her eighties.

And, as JLG notes in the comment above, "young woman" is often used to describe a woman who has not yet settled into adulthood. However, it can also be used to describe a teenager or even a child, and the line demarcating the lower age limit is pretty blurred. Note that the term young lady, when used as a form of direct address, is almost always in the context of a reproof of a teenager or younger, usually one's daughter or at least in one's care:

If you think you can go out before you clean your room, young lady, you've got another think coming.

Solution 3:

It's not gender-specific, but what's wrong with adolescent? Or the more informal teen? Granted, also not gender-specific.

Oh wait, you're talking about girls in their 20s. I think you're stuck with young woman.