Recently, the bias-related issues in GNN-based link prediction have raised widely spread concerns. In this paper, we emphasize the bias on links across different node clusters, which we call cross-links, after considering its significance in both easing information cocoons and preserving graph connectivity. Instead of following the objective-oriented mechanism in prior works with compromised utility, we empirically find that existing GNN models face severe data bias between internal-links (links within the same cluster) and cross-links, and this inspires us to rethink the bias issue on cross-links from a data perspective. Specifically, we design a simple yet effective twin-structure framework, which can be easily applied to most of GNNs to mitigate the bias as well as boost their utility in an end-to-end manner. The basic idea is to generate debiased node embeddings as demonstrations, and fuse them into the embeddings of original GNNs. In particular, we learn debiased node embeddings with the help of augmented supervision signals, and a novel dynamic training strategy is designed to effectively fuse debiased node embeddings with the original node embeddings. Experiments on three datasets with six common GNNs show that our framework can not only alleviate the bias between internal-links and cross-links, but also boost the overall accuracy. Comparisons with other state-of-the-art methods also verify the superiority of our method.