Hybrid organic-inorganic halide perovskites represent a promising next-generation photovoltaic material with drawbacks for structure stability and composition concerns. Demonstrations of ion migration and molecular dynamics suggest room for structural contraction and subsequent property adjustments. Here, we deploy dielec-tric and infrared spectroscopy under external pressure to probe the full structural phase diagram and dielectric response of methylammonium lead halide perovskites CH3NH3PbX3 (X = I, Br, or Cl). Ion migration can be fully suppressed by pressure beyond 4 GPa. The low-temperature orthorhombic phase transition can be gradually enhanced and stabilized at ambient conditions with increasing pressure. A slow relaxation mode, presumably the motion of the CH3NH+3 cation, is observed at lower pressure and is absent in the orthorhombic phase for every halide.