This paper considers the manner in which turbulent premixed flames respond to a superposition of turbulent and narrowband disturbances. This is an important fundamental problem that arises in most combustion applications, as turbulent flames exist in hydrodynamically unstable flow fields and/or in confined systems with narrowband acoustic waves. This paper presents the first measurements of the sensitivity of the turbulent displacement speed to harmonically oscillating flame wrinkles. The flame is attached to a transversely oscillating, heated wire, resulting in the introduction of coherent, convecting wrinkles on the flame. The approach flow turbulence is varied systematically using a variable turbulence generator, enabling quantification of the effect of turbulent flow disturbances on the harmonic wrinkles. Mie scattering measurements are used to quantify the flame edge dynamics, while high speed particle image velocimetry is used to measure the flow field characteristics. By ensemble averaging the results, the ensemble-averaged flame edge and flow characteristics are recovered. For low turbulence intensities, sharp cusps are present in the negative curvature regions of the ensemble-averaged flame position, similar to laminar flames. These cusps are smoothed out at high turbulence intensities. The coherent, ensemble-averaged flame wrinkle amplitude decays with increasing turbulence intensity and with downstream distance. In addition, the ensemble-averaged turbulent flame speed is modulated in space and time. The most significant result of these measurements is the clear demonstration of the correlation between the ensemble-averaged turbulent flame speed and ensemble-averaged flame curvature, with the phase-dependent flame speed increasing in regions of negative curvature. These results have important implications on turbulent combustion physics and modelling, since quasi-coherent velocity disturbances are nearly ubiquitous in shear driven, high turbulent flows and/or confined systems with acoustic feedback. Specifically, these data clearly show that nonlinear interactions occur between the multi-scale turbulent disturbances and the more narrowband disturbances associated with coherent structures. In other words, conceptual models of the controlling physics in combustors with shear driven turbulence must account for the fundamentally different effects of spectrally distributed turbulent disturbances and more narrowband, quasi-coherent disturbances.