We report on the observation of a complex nonlinear intensity dependence of the circular and linear photogalvanic currents induced by infrared radiation in compensated (Bi0.3Sb0.7)(2) (Te0.1Se0.9)(3) three-dimensional topological insulators. The photocurrents are induced by direct optical transitions between topological surface and bulk states. We show that an increase in the radiation intensity results first in a highly superlinear rise in the amplitude of both types of photocurrents, whereas at higher intensities the photocurrent saturates. Our analysis of the observed nonlinearities shows that the superlinear behavior of the photocurrents is caused by heating of the electron gas, while the saturation is induced by slow relaxation of the photoexcited carriers, resulting in absorbance bleaching. The observed nonlinearities give access to the Fermi-level position with respect to the Dirac point and the energy relaxation times of Dirac fermions, providing an experimental room-temperature probe of topological surface states.