Summary: We consider the Schrödinger operator \[ P = h^2 \Delta_g + V \] on \(\mathbb{R}^n\) equipped with a metric \(g\) that is Euclidean outside a compact set. The real-valued potential \(V\) is assumed to be compactly supported and smooth except at conormal singularities of order \(-1- \alpha\) along a compact hypersurface \(Y\). For \(\alpha >2\) (or even \(\alpha >1\) if the classical flow is unique), we show that if \(E_0\) is a non-trapping energy for the classical flow, then the operator \(P\) has no resonances in a region \[ [E_0 - \delta, E_0+ \delta] - i[0, \nu_0 h \log(1 /h ). \] The constant \(\nu_0\) is explicit in terms of \(\alpha\) and dynamical quantities. We also show that the size of this resonance-free region is optimal for the class of piecewise-smooth potentials on the line.