Cross-chain bridges have become essential infrastructure in the blockchain ecosystem, enabling interoperability across heterogeneous networks. However, externally verified bridges—those relying on validator committees—often expose validator identities and rely on strong trust assumptions, making them vulnerable to coercion, bribery, and targeted attacks. While prior work has improved validator reliability and decentralized trust distribution, validator anonymity remains an overlooked yet critical security concern. In this paper, we propose AnsBridge, a privacy-preserving cross-chain bridge that enables anonymous yet verifiable participation of validators. Validators register pseudonymously via a stake-based mechanism and prove ownership of secret keys through zero-knowledge proofs, without revealing their identities. Temporary public keys are added to a cryptographic accumulator, enabling anonymous eligibility validation. We further design protocols for efficient identity update and revocation using accumulator state updates and Bloom filters. We formally analyze the security of AnsBridge, demonstrating its resistance to validator-targeted attacks and its ability to preserve privacy without compromising correctness. We also implement a prototype and conduct performance evaluations, showing that AnsBridge achieves low-latency verification and compact proof sizes, making it suitable for real-world deployment. Our work highlights the importance of validator anonymity in cross-chain infrastructure and presents a practical design that balances privacy, decentralization, and efficiency.