  In this article, we study propagation of defect measures for Schr\"odinger
operators, $-h^2\Delta_g+V$, on a Riemannian manifold $(M,g)$ of dimension $n$
with $V$ having conormal singularities along a hypersurface $Y$ in the sense
that derivatives along vector fields tangent to $Y$ preserve the regularity of
$V$. We show that the standard propagation theorem holds for bicharacteristics
travelling transversally to the surface $Y$ whenever the potential is
absolutely continuous. Furthermore, even when bicharacteristics are tangent to
$Y$ at exactly first order, as long as the potential has an absolutely
continuous first derivative, standard propagation continues to hold.
