A simple and efficient screening model for studying the effects of superstrong magnetic fields (such as those of magnetars) on thermonuclear reaction rates on magnetar surfaces is proposed in this paper. The most interesting thermonuclear reactions, including hydrogen buing by the CNO cycle and helium buing by the triple alpha reaction, are investigated on the surface of magnetars. We find that the superstrong magnetic fields can increase the thermonuclear reaction rates by many orders of magnitude. The enhancement may have a dramatic effect on the thermonuclear runaways and bursts on the surfaces of magnetars.