Numerical modelling of dynamic rupture is conducted along faults separating similar and dissimilar materials. Supershear transition is enhanced in the direction of slip of the stiffer material (the negative direction) due to the bimaterial effect whereby a decrease in normal stress in front of the crack tip supports yielding ahead of the rupture. In the direction of slip of the more compliant material (the positive direction), an increase in normal stress ahead of the rupture tip delays or prevents the supershear transition, whereas the impact of the bimaterial effect on subshear ruptures is to promote rupture in the positive direction due to the tensile stress perturbation behind the rupture tip in this direction. We demonstrate that the material contrast and the parameter S control whether the transition from sub- to supershear velocity (supershear transition) is smooth or follows the Burridge–Andrews mechanism. Supershear transition along interfaces separating dissimilar materials is possible for higher values of the parameter S than supershear transition along material interfaces separating similar materials. The difference between pulse-like and crack-like rupture is small with regard to the supershear transition type.