Turbulent flows can exhibit extreme events, which are characterized by sudden bursts in the system observables. These events pose significant challenges for prediction and control owing to their intermittent, high-dimensional, and strongly nonlinear nature. We present a data-driven control framework for the suppression of extreme events in turbulent flows, leveraging reduced-order modeling for nonlinear compression of high-dimensional flow fields and a data-driven clustering algorithm for the identification of precursors to extreme events. A control law is defined in the low-dimensional latent space and optimized to efficiently mitigate extreme events through actuation values associated with the preidentified clusters. Since invariance transformations present in many nonlinear systems lead to an inflated latent space, we employ a symmetry-aware autoencoder to establish a compact latent space. By analyzing high-fidelity simulations of a canonical chaotic flow (2D Kolmogorov flow) whose extreme events correspond to short-lasting transition to high dissipation rate state, we demonstrate how the resulting closed-loop dynamics exhibit a substantial reduction in the frequency (ca. 99.4%) and intensity of extreme events. This is illustrated in the accompanying figure, which shows that the cases with control do not exhibit high dissipation rate states. The ability to incorporate controller and actuator latencies relevant to practical applications of the framework is further demonstrated. This study highlights the efficacy of data-driven methods that require little to no prior knowledge of the underlying system dynamics to achieve effective flow control, providing a pathway for general real-time suppression of extreme events in turbulent flows.