Tightly bound excitons in monolayer semiconductors represent a versatile platform to study two-dimensional propagation of neutral quasiparticles. Their intrinsic properties, however, can be severely obscured by spatial energy fluctuations due to a high sensitivity to the immediate environment. Here, we take advantage of the encapsulation of individual layers in hexagonal boron nitride to strongly suppress environmental disorder. Diffusion of excitons is then directly monitored using time and spatially resolved emission microscopy at ambient conditions. We consistently find very efficient propagation with linear diffusion coefficients up to 10 cm(2)/s, corresponding to room-temperature effective mobilities as high as 400 cm(2)/Vs as well as a correlation between rapid diffusion and short population lifetime. At elevated densities we detect distinct signatures of many-particle interactions and consequences of strongly suppressed Auger-type exciton-exciton annihilation. A combination of analytical and numerical theoretical approaches is employed to provide pathways toward comprehensive understanding of the observed linear and nonlinear propagation phenomena. We emphasize the role of dark exciton states and present a mechanism for diffusion facilitated by free-electron hole plasma from entropy-ionized excitons.