If you just want a reference, here is one:
Martin R. Bridson and James Howie, Normalizers in Limit Groups, Math. Ann. 337 (2007), no. 2, 385–394.
They prove that if $G$ is a limit group and $H$ is a nontrivial finitely generated subgroup then either the normalizer of $H$ in $G$ is abelian or $H$ has finite index in its normalizer. Thus, if $H$ is normal finitely generated subgroup in $G$ then either $H$ is of finite index in $G$ or $H$ is trivial or $G$ is abelian.
I will not give the definition of limit groups, but, according to the paper cited above: Examples of limit groups include all finitely generated free or free abelian groups, and the fundamental groups of all closed surfaces of Euler characteristic at most $−2$. The free product of finitely many limit groups is again a limit group, which leads to further examples.
Corollary. If $G$ is isomorphic to the fundamental group of a surface of genus $\ge 2$ then every finitely generated normal subgroup of $G$ is either trivial or has finite index in $G$.
Here is a hyperbolic geometry proof of this corollary, my main reference is
S. Katok, Fuchsian Groups, Chicago Lectures in Mathematics, University of Chicago Press, 1992.
First of all, if $S$ is a closed surface of genus $\ge 2$ then $S$ is homeomorphic to the quotient of the hyperbolic plane ${\mathbb H}^2$ by a discrete torsion-free subgroup $G< PSL(2, {\mathbb R})$ acting on the upper half plane ${\mathbb H}^2$ isometrically with respect to the hyperbolic metric. Groups $G$ as above are called Fuchsian groups (no requirement on compactness of the quotient space). (In general Fuchsian group are allowed to contain elements of finite order, but I will limit myself to torsion free groups only.) Fuchsian groups (rather unimaginatively) are divided in two classes: Groups of the 1st kind and groups of the 2nd kind.
Here are some facts about Fuchsian groups $G$:
If $Area({\mathbb H}^2/G)$ is finite then $G$ of the 1st kind.
If $Area({\mathbb H}^2/G)<\infty$ and $H$ is a subgroup of $G$ then
$[G:H]$ equals the ratio of areas:
$$
Area({\mathbb H}^2/H)/ Area({\mathbb H}^2/G).
$$
This follows from the fact that area of a surface is multiplicative with respect to degree $d$ coverings ($d$ could be infinite).
A nontrivial normal subgroup in a Fuchsian group of the 1st kind is again of the 1st kind, see this MSE question.
Each finitely generated Fuchsian group is geometrically finite, i.e. has a finitely sided fundamental polygon.
If $H$ is a geometrically finite Fuchsian group of the 1st kind then
$Area({\mathbb H}^2/H)<\infty$.
Using these 5 properties, here is a proof of the corollary:
Suppose that $G$ is a Fuchsian group such that ${\mathbb H}^2/G$ is compact. Hence, the area of the quotient is finite, hence, $G$ is of the 1st kind (Property 1). Suppose that $H< G$ is a nontrivial normal subgroup of infinite index. Hence, $Area({\mathbb H}^2/H)=\infty$ (Property 2).
Moreover, $H$ is of the 1st kind (Property 3). If $H$ were, in addition, finitely generated, it would be a geometrically finite subgroup of the 1st kind (Property 4), which implies that $Area({\mathbb H}^2/H)< \infty)$ (Property 5). A contradiction.