Cross-chain bridges have emerged as essential infrastructure within the blockchain ecosystem. However, as their usage grows, so does the risk of fraudulent and illicit activities, which pose significant challenges for manual monitoring and detection. Anomaly detection has become a critical mechanism for uncovering suspicious or irregular behaviors in such environments. This paper presents an investigation on anomaly detection aimed at identifying potentially illicit activities via cross-chain bridges. Specifically, we apply data analytics techniques to detect transactions involving addresses linked to crypto-mixer services. Our evaluation is conducted on a highly imbalanced dataset that reflects real-world cross-chain bridge activity. Using supervised learning models, we investigated the effectiveness of various detection strategies. We formulated the anomaly detection as binary classification and utilized five machine learning models (Random Forest, Gradient Boosting, Logistic Regression, Support Vector Machine, and XGBoost) in this study. The findings underscore that in security-critical environments such as cross-chain bridges, missing an abnormal transaction poses a greater risk than incorrectly flagging a legitimate one. Therefore, models in such contexts should prioritize high recall, even at the expense of increased false positives. This trade-off ensures comprehensive coverage of suspicious activities, with the understanding that flagged normal accounts may undergo manual or secondary verification.