The saturated state of turbulence driven by the ion-temperature-gradient instability is investigated using a two-dimensional long-wavelength fluid model that describes the perturbed electrostatic potential and perturbed ion temperature in a magnetic field with constant curvature (a Z-pinch) and an equilibrium temperature gradient. Numerical simulations reveal a well-defined transition between a finite-amplitude saturated state dominated by strong zonal-flow and zonal-temperature perturbations, and a blow-upstate that fails to saturate on a box-independent scale. We argue that this transition is equivalent to the Dimits’ transition from a low-transport to a high-transport state seen in gyro-kinetic numerical simulations (Dimits et al. 2000). A quasi-static, staircase-like structure of the temperature gradient intertwined with zonal flows, which have patch-wise constant shear, emerges near the Dimits’ threshold. The turbulent heat flux in the low-collisionality, near-marginal state is dominated by turbulent bursts, triggered by coherent long-lived structures closely resembling those found in gyro-kinetic simulations with imposed equilibrium flow shear (van Wyk et al. 2016). The break up of the low-transport, Dimits’ regime is linked to a competition between the two different sources of poloidal momentum in the system — the Reynolds stress and the advection of the diamagnetic flow by the E×B flow. By analysing the linear ITG modes, we obtain a semi-analytic model for the Dimits’ threshold at large collisionality.