Abelian Chern-Simons-Maxwell theory can emerge from the bosonization of the (2+1)-dimensional Thirring model that describes interacting Dirac fermions. Here we show how the Thirring model manifests itself in the low energy limit of a two-dimensional tight-binding model of spinless fermions. To establish that, we employ a modification of Haldane's model, where the "doubling" of fermions is rectified by adiabatic elimination. Subsequently, fermionic interactions are introduced that lead to the analytically tractable Thirring model. By local density measurements of the lattice fermions we can establish that for specific values of the couplings the model exhibits the confining (2+1)-dimensional QED phase or a topological ordered phase that corresponds to the Chern-Simons theory. The implementation of the model as well as the measurement protocol are accessible with the current technology of cold atoms in optical lattices.