Among the recent experimental studies focusing on the stochastic nature of bedload transport, some results are particularly intriguing: in experiments in which tagged particles (tracers) are tracked, the statistical moments of the time evolution of particle positions show anomalous scaling. Diffusion of tracer particles is described as super-diffusive on short time scales while on longer time scales, behavior is sub-diffusive, probably because the distribution of rest times is heavy tailed (as particles may be buried in the bed for relatively long times). We propose here a different perspective based on the statistics of bedload rate discharge. Experimental studies were conducted on a steep slope erodible bed made up of particles with a narrow size distribution. The flow conditions were typical of mountain rivers flows, characterized by a low submersion ratio and supercritical Froude numbers. Fluctuations of the solid discharge records reveal a self-similar behaviour over a wide range of time scales. Wherever this scaling holds, the solid discharge records exhibit a Hurst exponent approximately equal to 0.8. Consequently, bedload rate discharge is a self-similar stochastic process that lies in-between the random walk and white noise. This anomalous behavior is found to be mainly due to long range time correlations in the data series. Finally, we propose plausible physical explanations and their translation within a stochastic bedload transport model.