We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the L infinity and total variation bounds and viscosity. This answers a conjecture by Ayala and Protas (2011 Physica D 240 1553-63), based on numerical evidence, for the viscous Burgers equation.