This talk deals with the smoothing effect for entropy solutions of conservation laws with general nonlinear convex fluxes on $\mathbb{R}$. Beside convexity, no additional regularity is assumed on the flux. Thus, the well-known $BV$ smoothing effect for $C^2$ uniformly convex fluxes discovered independently by P. D. Lax and O. Oleinik is generalized for fluxes only locally Lipschitz. Therefore, the wave velocity can be discontinuous and the one-sided Oleinik inequality is lost. This inequality is usually the fundamental tool to get a sharp regularizing effect for the entropy solution. The wave velocity is modified in order to get an Oleinik inequality useful for a new modified wave front tracking algorithm. The unique entropy solution cannot be a function with bounded variation but belongs to a generalized BV space which depends only on the non-linearity of the flux.