  We consider negative moments of quadratic Dirichlet $L$--functions over
function fields. Summing over monic square-free polynomials of degree $2g+1$ in
$\mathbb{F}_q[x]$, we obtain an asymptotic formula for the $k^{\text{th}}$
shifted negative moment of $L(1/2+\beta,\chi_D)$, in certain ranges of $\beta$
(for example, when roughly $\beta \gg \log g/g $ and $k<1$). We also obtain
non-trivial upper bounds for the $k^{\text{th}}$ shifted negative moment when
$\log(1/\beta) \ll \log g$. Previously, almost sharp upper bounds were obtained
in \cite{ratios} in the range $\beta \gg g^{-\frac{1}{2k}+\epsilon}$.
