We experimentally investigate dissipation in mechanical resonators made of a disordered superconducting thin film of a Molybdenum-Rhenium(MoRe) alloy. Electrostatically driving the drum with a resonant AC voltage, we detect its motion using a superconducting microwave cavity. From the temperature dependence of mechanical resonance frequencies and quality factors, we find evidence for non-resonant, mechanically active two-level systems (TLSs) limiting its quality factor at low temperature. In addition, we observe a strong suppression of mechanical dissipation at large mechanical driving amplitudes, suggesting an unconventional saturation of the non-resonant TLSs. These observations shed light on the mechanism of mechanical damping in superconducting drums and routes towards understanding dissipation in such devices.