Simultaneous Speech Translation (SimulST) involves generating target language text while continuously processing streaming speech input, presenting significant real-time challenges. Multi-task learning is often employed to enhance SimulST performance but introduces optimization conflicts between primary and auxiliary tasks, potentially compromising overall efficiency. The existing model-level conflict resolution methods are not well-suited for this task which exacerbates inefficiencies and leads to high GPU memory consumption. To address these challenges, we propose a Modular Gradient Conflict Mitigation (MGCM) strategy that detects conflicts at a finer-grained modular level and resolves them utilizing gradient projection. Experimental results demonstrate that MGCM significantly improves SimulST performance, particularly under medium and high latency conditions, achieving a 0.68 BLEU score gain in offline tasks. Additionally, MGCM reduces GPU memory consumption by over 95\% compared to other conflict mitigation methods, establishing it as a robust solution for SimulST tasks.