Fluids subjected to suitable forcing will exhibit turbulence, with characteristics strongly affected by the fluid’s physical properties and dimensionality. In this work, we explore two-dimensional (2D) quantum turbulence in an oblate Bose-Einstein condensate confined to an annular trapping potential. Experimentally, we find conditions for which small-scale stirring of the condensate generates disordered 2D vortex distributions that dissipatively evolve toward persistent currents, indicating energy transport from small to large length scales. Simulations of the experiment reveal spontaneous clustering of same-circulation vortices and an incompressible energy spectrum with $k^{-5/3}$ dependence for low wave numbers $k$. This work links experimentally observed vortex dynamics with signatures of 2D turbulence in a compressible superfluid.