Bacterial biofilms have an enormous impact on medicine, industry and ecology. These microbial communities are generally considered to adhere to surfaces or interfaces. Nevertheless, suspended filamentous biofilms, or streamers, are frequently observed in natural ecosystems where they play crucial roles by enhancing transport of nutrients and retention of suspended particles. Recent studies in streamside flumes and laboratory flow cells have hypothesized a link with a turbulent flow environment. However, the coupling between the hydrodynamics and complex biofilm structures remains poorly understood. Here, we report the formation of biofilm streamers suspended in the middle plane of curved microchannels under conditions of laminar flow. Experiments with different mutant strains allow us to identify a link between the accumulation of extracellular matrix and the development of these structures. Numerical simulations of the flow in curved channels highlight the presence of a secondary vortical motion in the proximity of the corners, which suggests an underlying hydrodynamic mechanism responsible for the formation of the streamers. Our findings should be relevant to the design of all liquid-carrying systems where biofilms are potentially present and provide new insights on the origins of microbial streamers in natural and industrial environments.