The propulsive fins of ray-finned fish are used for large scale locomotion and fine maneuvering, yet also provide sensory feedback regarding hydrodynamic loading and the surrounding environment. This information is gathered via nerve cells in the webbing between their fin rays. A similar bioinspired system that can gather force feedback from fin motion could enable valuable insight into robotic underwater locomotion improving swimming efficiency and orientation. Fins are largely composed of bendable rays that support an elastic membranous web. In this investigation we have produced a stretch-sensing web that can be used as a sensor for a robotic fin; a proprioceptive fin webbing capable of measuring hydrodynamic loads. Our soft capacitive sensor web is embedded in 350 μm thin film that is held between wires which emulate fin rays. These sensor web constructs were successfully tested in water tunnels and maintained their sensory performance at speeds up to 0.7 m s^-1 and at angles-of-attack up to 90 degrees. We demonstrated sensor response as a function of water speed and angle of attack. Induced vibrations in the membrane from vortex shedding and flutter at high speeds were mitigated through the addition of passive chordwise stiffeners and tensioning of the membrane was investigated. Through understanding sensing membrane behavior in flow, the development of specialized fin webbing sensors becomes possible. This invention thus presents a milestone in advanced hydrodynamic sensing in fish robots enabling us to push further towards autonomous control loops in fish robots.