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This work combines experimental observations from the ASDEX Upgrade tokamak with related gyrokinetic simulations of the turbulence moving from L-mode toward, and beyond, the L–H transition. Dedicated experiments have been performed with slow steps of increasing electron cyclotron heating power. Gyrokinetic simulations of the edge turbulence of these plasmas highlight the key roles of the non-linear electromagnetic effects and the external flow shear (E × B shear), both related to the evolution of the plasma pressure profile with increasing heating power. The increase in the plasma βe destabilizes turbulence at low toroidal mode numbers, that, in turn, is strongly suppressed by the external flow shear. This allows the plasma pressure profiles to evolve without a sharp rise in the turbulent fluxes. When all the experimentally measured plasma parameters are consistently included as inputs of the local gyrokinetic simulations, both the experimental electron and ion heat fluxes are quantitatively reproduced on the whole L-mode phase of the selected discharge. Simulations carried out with edge parameters of an ELM-free H-mode phase still show the importance of the mechanisms discussed earlier while also indicating possible limitations of the local approach. [ABSTRACT FROM AUTHOR]