A turbulent flow mixes in general more rapidly a passive scalar than a laminar flow does. From an energetic point of view, for statistically homogeneous or periodic flows, the laminar regime is more efficient. However, the presence of walls may change this picture. We consider in this investigation mixing in two-dimensional laminar and turbulent wall-bounded flows using direct numerical simulation. We show that for sufficiently large Schmidt number, turbulent flows more efficiently mix a wall-bounded scalar field than a chaotic or laminar flow does. The mixing efficiency is shown to be a function of the Péclet number, and a phenomenological explanation yields a scaling law, consistent with the observations.