Actually, this is true in more generality:

- You don't need $f, X$, or $Y$ to be normal or projective (this is of course, trivial, they are all red herrings). You only need $g$ to be proper (I suppose you meant that $S$ is a projective surface, so that's covered).
- If all the fibers of $g$ are connected and
**any** fiber of $g$ is contracted, then all fibers are contracted. This is known as the "There are no bowties in algebraic geometry" theorem (or more conventionally called "Rigidity Lemma").
- From your formulation it seems that you are assuming that $C$ is irreducible. Because of the previous point you only need connected, but that is obviously necessary.

The proof is not too hard, and you should try. I believe this was first proved by Mumford when $S=F\times C$ and $g$ is the projection. A good place to look for the proof of this more general statement is Lemma 1.6 in Kollár-Mori98.