We determine the conditions for the occurrence of Turing instabilities in activator-inhibitor systems, where one component undergoes subdiffusion and the other normal diffusion. If the subdiffusing species has a nonlinear death rate, then coupling between the nonlinear kinetics and the memory effects of the non-Markovian transport process advances the Turing instability if the inhibitor subdiffuses and delays the Turing instability if the activator subdiffuses. We apply the results of our analysis to the Schnakenberg model, the Gray-Scott model, the Oregonator model of the Belousov-Zhabotinsky reaction, and the Lengyel-Epstein model of the chlorine dioxide-iodine-malonic acid reaction.