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. In this work we present a predominantly 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 is optimized to efficiently mitigate extreme events through actuation values associated with the preidentified clusters. To address issues in the latent space structure that arise from invariance transformations present in many nonlinear systems, leading to an inflated latent space, we employ symmetry-aware autoencoders to establish a more structured and compact latent space. By analyzing high-fidelity simulations of a canonical chaotic flow (Kolmogorov flow), we demonstrate how the resulting closed-loop dynamics exhibit a substantial reduction (up to ca. 99.4%) in the frequency and intensity of extreme events. The framework exhibits scalability to a more chaotic flow regime characterized by higher Reynolds number without encountering numerical instabilities, achieving a 96.9% reduction in the occurrence of extreme events. The ability to incorporate controller limitations, including actuator latency relevant to practical applications of the framework, has also been 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, thereby providing a pathway for general real-time suppression of extreme events in turbulent flows. ... Read more