The Kolmogorov theory of turbulence at high Reynolds numbers is based on the hypotheses of statistical isotropy of the flow and locality of the nonlinear interactions, when the velocity fluctuations in the inertial interval can be treated as inviscid. We present a new concept of generation of the Kolmogorov spectrum through the formation of folds for the vorticity field. Based on numerical simulations of the 3D incompressible Euler equations with generic large-scale initial conditions, we show that, unlike the Kolmogorov theory, the early turbulence turns out to be strongly anisotropic due to the formation of thin pancake structures of intense vorticity, corresponding to narrow jets in the energy spectrum. We establish, both numerically and analytically, the universal self-similar scaling for pancake structures, where the exponentially growing vorticity maximum is related to decreasing pancake width by the Kolmogorov-type power law, W_{max} ~ L^{-2/3}, resulting from the folding of vorticity field. Increasing number of such structures overlap at the intermediate scales and, after angle average, lead to formation of the Kolmogorov energy spectrum in a fully inviscid flow.