A stochastic data-driven reduced-order model applicable to a wide range of turbulent natural and engineering flows is presented. Combining ideas from Koopman theory and spectral model order reduction, the stochastic low-dimensional inflated convolutional Koopman model accurately forecasts short-time transient dynamics while preserving long-term statistical properties. A discrete Koopman operator is used to evolve convolutional coordinates that govern the temporal dynamics of spectral orthogonal modes, which, in turn, represent the energetically most salient large-scale coherent flow structures. Turbulence closure is achieved in two steps: first, by inflating the convolutional coordinates to incorporate nonlinear interactions between different scales, and second, by modelling the residual error as a stochastic source. An empirical dewhitening filter informed by the data is used to maintain the second-order flow statistics. The model uncertainty is quantified through either Monte–Carlo simulation or by directly propagating the model covariance matrix. The model is demonstrated on the Ginzburg–Landau equations, large-eddy simulation data of a turbulent jet, and particle image velocimetry data of the flow over an open cavity. In all cases, the model is predictive over time horizons indicated by a detailed error analysis and integrates stably over arbitrary time horizons, generating realistic surrogate data.