Premixed turbulent flames of methane-air stabilized on a Bunsen-type burner were studied to investigate the structure of the flame front at a wide range of turbulence intensities. The nondimensional turbulence rms velocity, rms velocity divided by the laminar flame speed, covered the range from about 3 to 24. The equivalence ratio was varied from 0.6 to stoichiometric. The flame front data were obtained using planar Rayleigh imaging, and particle image velocimetry was used to measure instantaneous velocity field for the experimental conditions studied. Flame front thickness increased slightly with increasing nondimensional turbulence rms velocity. There was no significant difference in flame thickening whether the flame thickness was evaluated at progress variable 0.5, corresponding to the reaction zone, or 0.3, corresponding to the preheat zone. Flame front curvature decreased with increasing turbulence rms velocity. Flame front curvature statistics displayed a Gaussian-like distribution, which centered about zero for all the flame conditions studied during the investigation. Flame surface densities evaluated from flame front images showed almost no dependence on the nondimensional turbulence intensity. Flame surface densities integrated over the flame brush volume also did not show any sensitivity to the nondimensional turbulence rms velocity. This was discussed in the framework of a flame surface density-based turbulent premixed flame propagation closure model. The implication is that the conceptual increase in flame surface density with turbulence may not be the dominant mechanism for flame velocity enhancement in turbulent combustion in the region specified as the flamelet combustion regime by the current turbulent premixed combustion diagrams. Further, the applicability of the flamelet approach may be limited to a much smaller range of conditions than presently believed.