High-throughput satellite communication systems are growing in strategic importance thanks to their role in delivering broadband services to mobile platforms and residences and/or businesses in rural and remote regions globally. Although precoding has emerged as a prominent technique to meet ever-increasing user demands, there is a lack of studies dealing with congestion control. This paper enhances the performance of multi-beam high throughput geostationary satellite systems under congestion, where the users' quality of service (QoS) demands cannot be fully satisfied with limited resources. In particular, we propose congestion control strategies, relying on simple power control schemes. We formulate a multi-objective optimization framework balancing the system sum-rate and the number of users satisfying their QoS requirements. Next, we propose two novel approaches that effectively handle the proposed multi-objective optimization problem. The former is a model-based approach that relies on the weighted sum method to enrich the number of satisfied users by solving a series of the sum-rate optimization problems in an iterative manner. The latter is a data-driven approach that offers a low-cost solution by utilizing supervised learning and exploiting the optimization structures as continuous mappings. The proposed general framework is evaluated for different linear precoding techniques, for which the low computational complexity algorithms are designed. Numerical results manifest that our proposed framework effectively handles the congestion issue and brings superior improvements of rate satisfaction to many users than previous works. Furthermore, the proposed algorithms show low run-time and make them realistic for practical systems.