  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 UV
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.
