Under stringent model type and variable distribution assumptions, score-based causal discovery methods learn the directed acyclic graph (DAG) from observational data by evaluating candidate graphs over an averaged score function. Despite the great success in low-dimensional linear systems, it has been observed that these approaches overly exploits easier-to-fit samples, thus inevitably learning spurious edges. Worse still, the common homogeneity assumption of most causal discovery methods can be easily violated due to the widespread existence of heterogeneous data in the real world, resulting in performance vulnerability when noise distributions vary. We propose a simple yet effective model-agnostic framework to boost causal discovery performance by dynamically learning the adaptive weights for the Reweighted Score function, ReScore for short, where the learned weights tailors quantitatively to the important degree of each samples. Intuitively, we leverage the bilevel optimization scheme to alternatively train a standard DAG learner first, then upweight the samples that the DAG learner fails to fit well and downweight the samples that the DAG learner easily extracts the causation information from. Extensive experiments on both synthetic and real-world datasets are carried out to validate the effectiveness of ReScore. We observe consistent and significant boosts in structure learning performance. We further visualize that ReScore concurrently mitigates the influence of spurious edges and generalizes to heterogeneous data. Finally, we perform theoretical analysis to guarantee the structure identifiability and the weight adaptive properties of ReScore. Our codes are available at https://github.com/anzhang314/ReScore.