Quantum particles on a lattice with competing long-range interactions are ubiquitous in physics; transition metal oxides(1,2), layered molecular crystals(3) and trapped-ion arrays(4) are a few examples. In the strongly interacting regime, these systems often show a rich variety of quantum many-body ground states that challenge theory(2). The emergence of transition metal dichalcogenide moire superlattices provides a highly controllable platform in which to study long-range electronic correlations(5-12). Here we report an observation of nearly two dozen correlated insulating states at fractional fillings of tungsten diselenide/tungsten disulfide moire superlattices. This finding is enabled by a new optical sensing technique that is based on the sensitivity to the dielectric environment of the exciton excited states in a single-layer semiconductor of tungsten diselenide. The cascade of insulating states shows an energy ordering that is nearly symmetric about a filling factor of half a particle per superlattice site. We propose a series of charge-ordered states at commensurate filling fractions that range from generalized Wigner crystals(7) to charge density waves. Our study lays the groundwork for using moire superlattices to simulate a wealth of quantum many-body problems that are described by the two-dimensional extended Hubbard model(3,13,14) or spin models with long-range charge-charge and exchange interactions(15,16). An optical sensing technique reveals an abundance of correlated insulating states at fractional fillings of moire superlattices that are proposed to arise from a series of charge-ordered states.