Propagation of turbulent premixed flamelets of finite thickness is considered using the model nonlinear equation proposed recently. The nonlinear equation takes into account both influence of external turbulence and intrinsic properties of a flame front caused by density variations across the front. The formula for the turbulent flame velocity is derived for the case of flamelets of small but finite thickness. The obtained results agree well with experimental data on flame propagation in a turbulent flow of moderate and high values of the Reynolds number. Known expressions for flame speed found by Clavin-Williams and by Yakhot for the case of zero thermal expansion of the fuel mixture are recovered.