We study a class of degenerate parabolic convection-diffusion equations, endowed with a mechanism for saturation of the diffusion flux, which corrects the unphysical gradient-flux relations at high gradients. This paper extends our previous works on the effects of diffusion with saturation on convection and the impact of saturation on porous media-type diffusion, where it has been demonstrated that a nonlinear saturating diffusion is susceptible to a self-induced formation of discontinuities. In this work we demonstrate that nonlinear convection enhances the breakdown effect. We carry both analytical and numerical studies of the model equation, ut + f(u)x = [φ(u)Q(ux, u)}x, where Q is a bounded increasing function, φ(0) = 0 and φ(u) ∼ un, n > 0 for u ∼ 0. Depending on a choice of n, we obtain two distinctive processes. If 0 ≤ n ≤ 1, a discontinuity forms only when the upstream-downstream disparity exceeds a critical threshold, but if n > 1, all travelling waves are found to have a sharp discontinuous front. In fact, given a compact or a semi-compact initial datum, the front will not start to move until such a discontinuity forms.