In this Letter we present turbulent flame speeds and their scaling from experimental measurements on constant-pressure, unity Lewis number expanding turbulent flames, propagating in nearly homogeneous isotropic turbulence in a dual-chamber, fan-stirred vessel. It is found that the normalized turbulent flame speed as a function of the average radius scales as a turbulent Reynolds number to the one-half power, where the average radius is the length scale and the thermal diffusivity is the transport property, thus showing self-similar propagation. Utilizing this dependence it is found that the turbulent flame speeds from the present expanding flames and those from the Bunsen geometry in the literature can be unified by a turbulent Reynolds number based on flame length scales using recent theoretical results obtained by spectral closure of the transformed G equation.