Structures of turbulent Bunsen flames in the corrugated-flamelet regime were investigated by use of a three-color six-beam laser-Doppler velocimetry system. Four different mixtures with identical laminar burning velocities (0.34 m/s) were selected to facilitate comparisons-lean and rich methane, and lean and rich propane. A bimodal distribution, not previously reported in the literature on turbulent Bunsen flames, was observed in the radial component of gas velocity off-axis in the turbulent flame brush. Our previous measurements enabled the low-velocity-mode to be identified as velocity fluctuations of the unburned mixture and the high-velocity mode as fluctuations of the burned-gas radial velocity. Favre-averaged and Reynolds-averaged reaction-progress variables were then calculated from these bimodal distributions, identifying an initially unexpected region near the flame tips where, at a fixed radius, the average progress variable decreased (rather than increasing) with increasing height over a short distance, likely through enhanced flamelet flapping, which has not been predicted by modeling but which appears to occur quite generally for sufficiently tall turbulent Bunsen flames in quiescent ambient environments, for the corrugated-flamelet regime. Conditioned and unconditioned Favre-average velocity components and intensities also were calculated from the data for future tests of modeling. The distributions of the progress variables also clearly showed that the turbulent burning velocity of the rich propane flame was appreciably larger than that of any of the other three, as was its radial flame-brush thickness at any given height, and its high-radial-velocity mode had a higher average velocity magnitude than the others. Similarly, the turbulent burning velocity and flame-brush thickness appear to be smaller for the lean propane flame. These differences can be attributed to influences of preferential oxygen diffusion to turbulence-induced flamelet bulges, not included in existing modeling approaches, for the rich propane flames, and to a corresponding inhibition of fuel diffusion to the bulges in the lean flames. The former phenomenon is related to but different from the well-known cellular-flamelet instability, these effects occurring for flames that are stable to diffusive-thermal disturbances. It was concluded that a greater fraction of the total amount of heat release occurs in the upstream half of the turbulent flame brush in the rich propane flame, producing enhanced flow divergence in the upstream region, while the reduced ability of the slowly diffusing fuel to reach bulges in the lean flame generates the opposite effect. The results point to directions in which turbulent-combustion modeling needs to be improved, and an approach to modeling this type of preferential-diffusion effect is suggested.