Curli are functional amyloids produced by proteobacteria like Escherichia coli as part of the extracellular matrix that holds cells together into biofilms. The molecular events that occur during curli nucleation and fiber extension remain largely unknown. Combining observations from curli amyloidogenesis in bulk solutions with real-time in situ nanoscopic imaging at the single-fiber level, we show that curli display polar growth, and we detect two kinetic regimes of fiber elongation. Single fibers exhibit stop-and-go dynamics characterized by bursts of steady-state growth alternated with periods of stagnation. At high subunit concentrations, fibers show constant, unperturbed burst growth. Curli follow a one-step nucleation process in which monomers contemporaneously fold and oligomerize into minimal fiber units that have growth characteristics identical to those of the mature fibrils. Kinetic data and interaction studies of curli fibrillation in the presence of the natural inhibitor CsgC show that the inhibitor binds curli fibers and predominantly acts at the level of fiber elongation.