We consider strong approximations of 1+1-dimensional stochastic PDEs driven by additive space-time white noise. It has been long proposed (Davie-Gaines '01, Jentzen-Kloeden '08), as well as observed in simulations, that approximation schemes based on samples from the stochastic convolution, rather than from increments of the underlying Wiener processes, should achieve significantly higher convergence rates with respect to the temporal timestep. For a large class of nonlinearities, with possibly superlinear growth, we prove a temporal rate of (almost) 1, a major improvement on the rate 1/4 that is known to be optimal for schemes based on Wiener increments. The spatial rate remains (almost) 1/2 as it is standard in the literature. The talk is based on a joint work with Ana Djurdjevac and Máté Gerencsér.