We consider the exponential Euler approximation of nonlinear stochastic reaction diffusion equations driven by a 1+1-dimensional white noise and prove a strong rate of convergence of (almost) 1/2 in space and, crucially, 1 in time, under a C^2_b-assumption on the nonlinearity. This generalizes the results from Jentzen/Kloeden '08 to truly nonlinear coefficients and thus overcomes the previous order barrier for schemes, that consider a semi-discrete noise term. In a second part, we generalize our techniques based on the stochastic sewing lemma to splitting schemes of SPDEs with one-sided Lipschitz nonlinearities. In particular, we combine our previous techniques with a clever buckeling, that avoids the use of exponential bounds (that may not be available) to moreover overcome the order barrier for SPDEs with superlinearily growing non-linearities. We remark that our techniques are very general in the sense, that they apply for all numerical schemes, for which apriori estimates can be derived. The talk is based on a joint work with Ana Djurdjevac and Máté Gerencsér.