It has been realized only recently that TeV emission from blazars can significantly heat the intergalactic medium (IGM) by pair-producing high-energy electrons and positrons, which in turn excite vigorous plasma instabilities, leading to a local dissipation of the pairs’ kinetic energy. In this work, we use cosmological hydrodynamical simulations to model the impact of this blazar heating on the Lyman α forest at intermediate redshifts (z∼ 2–3). We find that blazar heating produces an inverted temperature–density relation in the IGM and naturally resolves many of the problems present in previous simulations of the forest that included photoionization heating alone. In particular, our simulations with blazar heating simultaneously reproduce the observed effective optical depth and temperature as a function of redshift, the observed probability distribution functions (PDFs) of the transmitted flux, and the observed flux power spectra, over the full redshift range 2 < z < 3 analysed here. Additionally, by deblending the absorption features of Lyman α spectra into a sum of thermally broadened individual lines, we find superb agreement with the observed lower cut-off of the linewidth distribution and abundances of neutral hydrogen column densities per unit redshift. Using the most recent constraints on the cosmic ultraviolet (UV) background, this excellent agreement with observations does not require rescaling the amplitude of the UV background – a procedure that was routinely used in the past to match the observed level of transmitted flux. We also show that our blazar-heated model matches the data better than standard simulations even when such a rescaling is allowed. This concordance between Lyman α data and simulation results, which are based on the most recent cosmological parameters, also suggests that the inclusion of blazar heating alleviates previous tensions on constraints for σ8 derived from Lyman α measurements and other cosmological data. Finally, we show that blazar heating dramatically alters the volume-weighted temperature PDF, implying an important change in the strengths of structure formation shocks (and thereby possibly particle acceleration in these shocks). The density PDF is also modified, suggesting that blazar heating may have interesting effects on structure formation, particularly on the smallest galaxies.