We present a Vlasov, i.e. a kinetic Eulerian simulation study of nonlinear collisionless ion-acoustic shocks and solitons excited by an intense laser interacting with an overdense plasma. The use of the Vlasov code avoids problems with low particle statistics and allows a validation of particle-in-cell results. A simple, original correction to the splitting method for the numerical integration of the Vlasov equation has been implemented in order to ensure the charge conservation in the relativistic regime. We show that the ion distribution is affected by the development of a turbulence driven by the relativistic 'fast' electron bunches generated at the laser-plasma interaction surface. This leads to the onset of ion reflection at the shock front in an initially cold plasma where only soliton solutions without ion reflection are expected to propagate. We give a simple analytical model to describe the onset of the turbulence as a nonlinear coupling of the ion density with the fast electron currents, taking the pulsed nature of the relativistic electron bunches into account.