We demonstrate precise control of charged particle bunch shape with a cold atom electron and ion source to create bunches with linear and, therefore, reversible Coulomb expansion. Using ultracold charged particles enables detailed observation of space-charge effects without loss of information from thermal diffusion, unambiguously demonstrating that shaping in three dimensions can result in a marked reduction of Coulomb-driven emittance growth. We show that the emittance growth suppression is accompanied by an increase in bunch focusability and brightness, improvements necessary for the development of sources capable of coherent single-shot ultrafast electron diffraction of noncrystalline objects, with applications ranging from femtosecond chemistry to materials science and rational drug design.