•A new physical scenario of the influence of turbulence on reaction waves is studied.•A relevant theory is developed.•The theory is validated using direct numerical simulation data.•The theory explains recent experimental and direct numerical simulation data. A big database (23 cases characterized by Damköhler number less than unity) created recently in 3D Direct Numerical Simulation (DNS) of propagation of a statistically one-dimensional and planar, dynamically passive reaction wave in statistically stationary, homogeneous, isotropic turbulence is analyzed. On the one hand, the DNS data well support the classical Damköhler expression, i.e., square-root dependence of a ratio of turbulent and laminar consumption velocities on the turbulent Reynolds number. On the other hand, contrary to the common interpretation of the Damköhler theory and, in particular, to the concept of distributed burning, the DNS data show that the reaction is still localized to thin zones even at Da as low as 0.01, with the aforementioned ratio of the consumption velocities being mainly controlled by the reaction-zone-surface area. To reconcile these apparently inconsistent numerical findings, an alternative regime of propagation of reaction waves in a highly turbulent medium is analyzed, i.e., propagation of an infinitely thin reaction sheet is theoretically studied, with molecular mixing of the reactant and product being allowed in wide layers. In this limiting case, an increase in the consumption velocity by turbulence is solely controlled by an increase in the reaction-sheet area. Based on physical reasoning and estimates, the area is hypothesized to be close to the mean area of an inert iso-scalar surface at the same turbulent Reynolds number. This hypothesis leads to the aforementioned square-root dependence. Thus, both the DNS data and the developed theory show that a widely accepted hypothesis on penetration of small-scale turbulent eddies into reaction zones is not necessary to obtain the classical Damköhler scaling for turbulent consumption velocity.