The vast majority of bacteria live in sessile biofilms that colonize the channels, pores and crevices of confined structures. Flow in these structures carries the nutrients necessary for growth and generates stresses and detachment from surfaces. Conversely, bacteria tend to occupy a large part of the available space and, in so doing, increase resistance to flow and modify transport properties. Although the importance of advective transport and hydrodynamic forces on bacteria is well known, the complex feedback effects that control development in confined geometries are much less understood. Here, we study how couplings between flow and bacterial development control the spatio-temporal dynamics of Pseudomonas aeruginosa in microchannel flows. We demonstrate that nutrient limitation drives the longitudinal distribution of biomass, while a competition between growth and flow-induced detachment controls the maximum clogging and the temporal dynamics. We find that successive cycles of sloughing and growth cause persistent fluctuations of the hydraulic resistance and prevent the system from ever reaching a true steady-state. Our results indicate that these self-sustained fluctuations are a signature effect of biofilm development in confined flows and could thus be a key component of the spreading of biofilms in infections, environmental processes and engineering applications. Consistent with the description of other bursting events, such as earthquakes or avalanches, we further show that the dynamics of sloughing can be described as a jump stochastic process with a gamma distribution of interevent times. This stochastic modeling approach opens the way towards a new quantitative approach to the characterization of the apparent randomness and irreproducibility of biofilm experiments in such systems.