I am probably having a lot of confusion with the terminologies in shafarevich.
In page 131, Normal varieties, it states a corollary.
An irreducible algebraic curve is birational to a nonsingular projective curve.
Now I can't find "algebraic curve" defined in the book, as well as "algebraic variety". I guess algebraic variety = quasiprojective variety and algebraic curve = quasiprojective variety of dimension 1.
But to prove the corollary it wants us to use the Theorem 2.23, which states that
The normalization of a projective curve is projective.
Now I don't see how does that theorem apply to the corollary, as our "algebraic curve" need not be projective. (Anyway, I understand what is going on, i.e normal and non-singular coincides in dimension 1, but the confusion remains. )
I think your definitions are correct: for Shafarevich, "algebraic variety" always means "quasiprojective variety", and "curve" means "variety of dimension 1".
To answer your question, the point is that any quasi-projective algebraic curve $C$ is birational to a projective curve $\tilde{C}$ in a simple way: just take $\tilde{C}$ to be the closure of $C$ inside projective space. Then you can normalise $\tilde{C}$ to get something nonsingular and projective, and still birational to $C$.