To address the difficulty of predicting dynamic obstacle states and the increased decision uncertainty caused by incomplete perception and sensor noise in complex dynamic environments, this paper proposes a forward-dynamics-assisted asymmetric reinforcement learning framework for autonomous navigation, termed forward-dynamics-assisted asymmetric navigation (FANav). First, an environment forward dynamics model (E-FDM) is developed to learn the short-term state evolution of robot-environment interactions during training. The predicted environmental changes and collision risks are incorporated into policy optimization to enable anticipatory decision-making. During deployment, the E-FDM provides online short-horizon environmental predictions from local observations and the current action to support real-time obstacle avoidance. Second, to mitigate safety risks caused by prediction errors and sensor noise, a control barrier function (CBF)-based safety filter with short-term risk prediction is introduced at the execution layer. It adaptively adjusts the safety constraint strength using anticipatory risk information and minimally modifies the policy output through quadratic programming, thereby enhancing formal safety enforcement. Finally, an asymmetric proximal policy optimization (PPO) framework is adopted, in which privileged information, including forward predictions, is used to optimize the value network and improve policy learning efficiency and training stability. Experimental results show that the FANav achieves superior obstacle avoidance safety, decision robustness, and motion smoothness compared with existing baselines.