We report here the results of experiments on two flows – one on a convex surface and the other on a flat surface – designed to bring out explicitly the influence of streamwise curvature on relaminarization in highly favourable pressure gradients. In both flows, the initial conditions and the streamwise distribution of the Launder pressure-gradient parameter $K$ are virtually identical. The maximum value of $K$ is $6.2\,{\times}\, 10^{-6}$, well above the critical value of about $3.5\,{\times}\, 10^{-6}$ usually advocated for relaminarization. The spatial extent of the acceleration zone is of order 10 initial boundary-layer thicknesses, appreciably shorter than in earlier work in order better to simulate conditions at the leading edge of a typical aircraft wing. The fall in skin friction coefficient is steeper and the rise in shape factor sharper on the convex surface than on the flat surface, indicating that relaminarization on the convex surface is both more rapid and more nearly complete. In the crucial relaminarizing zone, two-layer quasi-laminar theory is found to predict the convex-surface mean-flow parameters more accurately than the flat-surface flow, without any explicit modelling of curvature effects. Thus, experimental results and supporting calculations both indicate that the dominant effect of streamwise convex curvature on the mean flow is to promote more rapid and complete relaminarization in an accelerated turbulent boundary layer, thus enhancing the probability of its occurrence on the leading edge of swept wings where both factors are significantly in operation.