A Multilayered Satellite Network (MLSN) tends to be a promising architecture in facilitating global ubiquitous broadband communication. However, unbalanced traffic distribution among its satellite layers should frequently occur, where the lower layers could get relatively congested while the upper layers remain underutilized. This unfair distribution of network traffic can lead to large end-to-end delay and severe throughput degradation. To cope with the above issue, we propose a Tailored Load-Aware Routing (TLAR) strategy to optimally distribute traffic load among the multiple satellite layers, so that the overall traffic congestion in the MLSN is minimized. In TLAR, an optimal portion of network load, which is decided based upon the newly arrived traffic estimation and theoretical analysis of traffic congestion rate in each layer, is detoured through the upper layer. The performance of the proposed routing method has been validated through extensive simulations, which demonstrate that TLAR can significantly alleviate traffic congestion, achieve low end-to-end delay and sustain improved throughput.