  The emergence of moir\'e materials with flat bands provides a platform to
systematically investigate and precisely control correlated electronic phases.
Here, we report local electronic compressibility measurements of a twisted
WSe$_2$/MoSe$_2$ heterobilayer which reveal a rich phase diagram of
interpenetrating Hofstadter states and electron solids. We show that this
reflects the presence of both flat and dispersive moir\'e bands whose relative
energies, and therefore occupations, are tuned by density and magnetic field.
At low densities, competition between moir\'e bands leads to a transition from
commensurate arrangements of singlets at doubly occupied sites to triplet
configurations at high fields. Hofstadter states (i.e., Chern insulators) are
generally favored at high densities as dispersive bands are populated, but are
suppressed by an intervening region of reentrant charge-ordered states in which
holes originating from multiple bands cooperatively crystallize. Our results
reveal the key microscopic ingredients that favor distinct correlated ground
states in semiconductor moir\'e systems, and they demonstrate an emergent
lattice model system in which both interactions and band dispersion can be
experimentally controlled.
