Classical nonlinear waves exhibit a phenomenon of condensation that results from the natural irreversible process of thermalization, in analogy with the quantum Bose-Einstein condensation. Wave condensation originates in the divergence of the thermodynamic equilibrium Rayleigh-Jeans distribution, which is responsible for the macroscopic population of the fundamental mode of the system. However, achieving complete thermalization and condensation of incoherent waves through nonlinear optical propagation is known to require prohibitive large interaction lengths. Here, we derive a discrete kinetic equation describing the nonequilibrium evolution of the random wave in the presence of a structural disorder of the medium. Our theory reveals that a weak disorder accelerates the rate of thermalization and condensation by several order of magnitudes. Such a counterintu-itive dramatic acceleration of condensation can provide an explanation for the recently discovered phenomenon of optical beam self-cleaning. We report experiments in multimode optical fibers that highlight the transition from an incoherent thermal distribution to wave condensation, with a con-densate fraction of up to 60% in the fundamental mode of the waveguide trapping potential.