Propagation of premixed turbulent flames is examined using a hybrid Navier Stokes/front tracking methodology, within the context of a hydrodynamic model. The flame, treated as a surface of density discontinuity separating the burned and unburned gases, propagates relative to the fresh mixture at a speed that depends on the local mixture (through a Markstein length) and flow conditions (through the stretch rate), and the flow field is modified in turn by gas expansion; only positive Markstein length are considered, where thermo-diffusive instabilities are absent. Depending on the Markstein length, we have identified in a previous publication two modes of propagation sub-critical and super-critical, based on whether the effects of the Darrieus Landau instability are absent or dominant, respectively. The results were limited to low turbulence intensities where the mathematical representation of the flame front was based on an explicit single-valued function. In the present paper we utilize a generalized representation of the flame surface that allows for multivalued and disjointed interfaces, thus extending the results to higher turbulence intensities. We show that when increasing the turbulence intensity the influence of the Darrieus Landau instability on the super-critical mode of propagation progressively decreases and in the newly identified highly-turbulent regime the flame is dominated completely by the turbulence for all values of Markstein numbers; i.e., with no distinction between sub- and super-critical conditions. Primary importance is given to the determination of the turbulent flame speed and its dependence on turbulence intensity which, when increasing the turbulence level, transitions from a quadratic to a sub-linear scaling. Moreover, the exponent of the sub-linear scaling for the turbulent flame speed is generally lower than the corresponding exponent for the scaling of the flame surface area ratio, which is often used for experimentally determining the turbulent flame speed. We show that the leveling in the rate of increase of the turbulent flame speed with turbulence intensity, is due to frequent flame folding and detachment of pockets of unburned gas that cause a reduction in the average main surface area of the flame, while the lower exponents in the scaling law for the turbulent flame speed compared to that of the flame surface area ratio is due to flame stretching. Disregarding the effect of flame stretch for mixtures of positive Markstein length results in overestimating the turbulent flame speed. Finally, we characterize the flame turbulence interaction via quantities such as the mean vorticity and mean strain, illustrating the effects of incoming turbulence on the flame and the modification of the flow by the flame on the unburned and burned sides. (C) 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.