In modern Software-Defined Networking (SDN), traffic engineering often faces critical challenges in maintaining Quality of Service (QoS) due to congestion over bottleneck links. Existing traffic engineering (TE) schemes frequently overlook the performance degradation caused by such bottlenecks, resulting in inefficient routing and poor load balancing. To address this issue, we model the traffic allocation problem as a bottleneck game with splittable flows, where users compete for shared bottleneck resources and select routes based on a link pricing strategy. We propose a distributed routing algorithm based on bottleneck game theory (DRB), which introduces a pricing application-driven network routing mechanism between users and the SDN controller. The controller periodically monitors link utilization, updates link prices accordingly, and broadcasts them to users. Each user independently selects the lowest-cost path, implicitly guided by bottleneck usage. This mechanism effectively steers traffic away from congested paths, minimizes queuing delay, and achieves global load balancing, thereby ensuring improved QoS. Extensive simulations on real-world topologies from the SNDlib dataset, including Abilene and GEANT, validate the effectiveness of DRB. Compared to state-of-the-art methods such as Hedera and ECMP, the proposed DRB algorithm consistently achieves higher throughput, lower latency, reduced jitter, and minimized packet loss under varying traffic demand levels. These results demonstrate that DRB effectively mitigates bottleneck-induced congestion and enhances QoS in SDN-based networks.