Taxiway routing and on-surface conflict avoidance are coupled safety-critical decision problems in airport surface operations. Existing planning and optimization methods are often limited by online computational cost, while reinforcement learning methods may struggle to represent downstream traffic conflicts and balance multiple objectives. This paper presents Conflict-aware Taxiway Routing (CaTR), a reinforcement learning framework for real-time multi-aircraft taxiway routing. CaTR constructs a grid-based airport surface environment with action masking, introduces a hierarchical foresight traffic representation to encode current and downstream conflict-related traffic conditions, and adopts a value-decomposed reinforcement learning strategy to prioritize sparse but safety-critical objectives. Experiments are conducted on a realistic environment based on Changsha Huanghua International Airport under multiple traffic density levels. Results show that CaTR achieves better safety--efficiency trade-offs than representative planning, optimization, and reinforcement learning baselines while maintaining practical runtime.