The complex dynamics of high-dimensional oscillatory flows can be simplified using phase-reduction analysis, providing a deeper understanding of the flow response to external perturbations. Although phase-based modeling and analysis have been utilized in recent studies on oscillatory fluid flows, their usage is often limited to single-frequency flows due to difficulties in addressing chaotic characteristics, emerging from multiple dominant frequencies with broadband spectra in turbulent flows. In order to overcome this limitation, we propose a data-driven framework that models the dynamics of turbulent flows with multiple dominant frequencies based on a set of oscillators. The representative oscillators are extracted from the flow field data by training specially designed autoencoders. The oscillator dynamics are modeled through a machine-learning technique using neural networks to accurately predict the multi-frequency oscillatory behavior of turbulent flows. We verify the oscillator-based model of the oscillatory turbulent flows by applying the proposed data-driven method to the three-dimensional supersonic turbulent flow over a cavity. We show that the oscillators extracted from the spanwise-averaged pressure field represent the dominant large-scale flow features and reflect the physical characteristics of the turbulent cavity flow. The data-driven oscillator dynamics model with observation-based correction accurately forecasts the oscillatory behavior of the turbulent cavity flow for a long period. The proposed data-driven method for reduced-order modeling of turbulent flows with oscillators will enable deeper investigations of perturbation dynamics and control of turbulent flows.