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ABSTRACT We explore the evolution of cold streams from the cosmic web that feed galaxies through their shock-heated circumgalactic medium (CGM) at cosmic noon, $$z\simeq 1-5$$. In addition to the hydrodynamical instabilities and radiative cooling that we have incorporated in earlier works, we embed the stream and the hot CGM in the gravitational potential of the host dark matter halo, deriving equilibrium profiles for both. Self-gravity within the stream is tentatively ignored. We find that the cold streams gradually entrain a large mass of initially hot CGM gas that cools in the mixing layer and condenses onto the stream. This entrainment, combined with the acceleration down the gravitational potential well, typically triples the inward cold inflow rate into the central galaxy, compared to the original rate at the virial radius, which makes the entrained gas the dominant source of gas supply to the galaxy. The potential sources for the hot gas to be entrained are recycled enriched gas that has been previously ejected from the galaxy, and fresh virial-shock-heated gas that has accumulated in the CGM. This can naturally elevate the star formation rate in the galaxy by a factor of $$\sim 3$$ compared to the gas accretion rate onto the halo, thus explaining the otherwise puzzling observed excess of star formation at cosmic noon. When accounting for self-shielding of dense gas from the ultraviolet background, we find that the energy radiated from the streams, originating predominantly from the cooling of the entrained gas, is consistent with observed Lyman-$$\alpha$$ blobs around galaxies.