  We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet
$L$--functions over $\mathbb{F}_q[x]$, as the base field $\mathbb{F}_q$ is
fixed and the genus of the family goes to infinity. According to conjectures of
Andrade and Keating, we expect the fourth moment to be asymptotic to $q^{2g+1}
P(2g+1)$ up to an error of size $o(q^{2g+1})$, where $P$ is a polynomial of
degree $10$ with explicit coefficients. We prove an asymptotic formula with the
leading three terms, which agrees with the conjectured result.
