  Let $\Sigma$ be a surface of negative Euler characteristic and $S$ a
generating set for $\pi_1(\Sigma,p)$ consisting of simple loops that are
pairwise disjoint (except at $p$). We show that the word length with respect to
$S$ of an element of $\pi_1(\Sigma,p)$ is given by its intersection number with
a well-chosen collection of curves and arcs on $\Sigma$. The same holds for the
word length of (a free homotopy class of) an immersed curve on $\Sigma$. As a
consequence, we obtain the asymptotic growth of the number of immersed curves
of bounded word length, as the length grows, in each mapping class group orbit.
