Through molecular dynamics simulation we demonstrate that. the time scale for local solvent reorganizations, which may control solute dynamics, can become very long for low- and near-critical-density supercritical fluids. We show that this can be ascribed to two precesses, the first being due to direct, interparticle potential interactions, and the second, which dominates on all but the shortest length scales, being due to a coupling between the dynamics of the local solvent environment and those of the long-length-scale density fluctuations which are present in compressible supercritical fluids. Specifically, as the critical point is approached and the correlation length of these density inhomogeneities increases, we find that the associated slowing of these extended collective fluctuations, known as "critical slowing down", generates a concomitant slowing of the local reorganization time.