We report results from an experimental investigation on the fluid-structure interactions of flapping foils with tapered thickness profiles actuated in a quiescent viscous fluid. We seek to assess the propulsive performance of two sets of flapping foils; one with a fixed average bending stiffness, the other one with a fixed mass ratio. We find that foils that are stiffer towards the root than at their tip produce higher values of thrust and efficiency simultaneously, over a wide range of driving frequencies. Our kinematic analysis reveals that more tapered foils naturally develop a traveling-wave-dominated motion. We perform particle image velocimetry to relate the dynamics and kinematics of the flapping foils to the dynamics of the surrounding fluid. For more tapered foils, we observe a stronger vorticity production and a wake pattern with enhanced downstream speed of the fluid. Our paper provides experimental evidence that tapered stiffness distributions robustly enhances propulsive performance.