The flow behavior of fiber suspensions has been studied extensively, especially in the limit of dilute concentrations and rigid fibers; at the other extreme, however, where the suspensions are concentrated and the fibers are highly flexible, much less is understood about the flow properties. We use a microfluidic method to produce uniform concentrated suspensions of high aspect ratio, flexible microfibers, and we demonstrate the shear thickening and gelling behavior of such microfiber suspensions, which, to the best of our knowledge, has not been reported previously. By rheological means, we show that flowing the suspension triggers the irreversible formation of topological entanglements of the fibers resulting in an entangled water-filled network. This phenomenon suggests that flexible fiber suspensions can be exploited to produce a new family of flow-induced gelled materials, such as porous hydrogels. A significant consequence of these flow properties is that the microfiber suspension is injectable through a needle, from which it can be extruded directly as a hydrogel without any chemical reactions or further treatments. Additionally, we show that this fiber hydrogel is a soft, viscoelastic, yield-stress material.