  We use the $q$-analogue of van der Corput's method to estimate short
character sums to smooth moduli. If $\chi$ is a primitive Dirichlet character
modulo a squarefree, $q^\delta$-smooth integer $q$ we show that
$$L(\frac12,\chi)\ll_\epsilon q^{\frac{27}{164}+O(\delta)+\epsilon}.$$
