We analyse the evolution of turbulence and gravitational instability of a galactic disc in a quasi-steady state governed by cosmological inflow. We focus on the possibility that the coupling between the in-streaming gas and the disc is maximal, e.g. via dense clumps, and ask whether the streams could be the driver of turbulence in an unstable disc with a Toomre parameter Q ∼ 1. Our fiducial model assumes an efficiency of ∼0.5 per dynamical time for the decay of turbulence energy, and ∼0.02 for each of the processes that depletes the disc gas, i.e. star formation, outflow and inflow within the disc into a central bulge. In this case, the in-streaming drives a ratio of turbulent to rotation velocity σ/V ∼ 0.2–0.3, which at z ∼ 2 induces an instability with Q ∼ 1, both as observed. However, in conflict with observations, this model predicts that σ/V remains constant with time, independent of the cosmological accretion rate, because mass and turbulence have the same external source. Such strongly coupled cosmological inflow tends to stabilize the disc at low z, with Q ∼ a few, which may be consistent with observations. The instability could instead be maintained for longer, with a properly declining σ/V, if it is self-regulated to oscillations about Q ≈ 1 by a duty cycle for disc depletion. However, the ‘off’ phases of this duty cycle become long at low z, which may be hard to reconcile with observations. Alternatively, the coupling between the in-streaming gas and the disc may weaken in time, reflecting an evolving nature of the accretion. If, instead, that coupling is weak at all times, the likely energy source for self-regulated stirring up of the turbulence is the inflow within the disc down the potential gradient (studied in a companion paper).