We study the effects of bismuth doping on the crystal structure and phase transitions in single crystals of the perovskite semiconductor methylammonium lead tribromide, MAPbBr3. By measuring the temperature-dependent specific heat capacity (Cp), we find that as the Bi doping increases, the phase transition assigned to the cubic to tetragonal phase boundary decreases in temperature. Furthermore, after doping we observe one phase transition between 135 and 155 K, in contrast to two transitions observed in the undoped single crystal. These results appear strikingly similar to previously reported effects of mechanical pressure on perovskite crystal structure. Using X-ray diffraction, we show that the lattice constant decreases as Bi is incorporated into the crystal, as predicted by density functional theory. We propose that bismuth substitutional doping on the lead site is dominant, resulting in BiPb+ centers that induce compressive chemical strain that alters the crystalline phase transitions.