Data theft and leakage, caused by external adversaries and insiders, demonstrate the need for protecting user data. Trusted Execution Environments (TEEs) offer a promising solution by creating secure environments that protect data and code from such threats. The rise of confidential computing on cloud platforms facilitates the deployment of TEE-enabled server applications, which are expected to be widely adopted in web services such as privacy-preserving LLM inference and secure data logging. One key feature is Remote Attestation (RA), which enables integrity verification of a TEE. However, $\textit{compatibility}$ issues with RA verification arise as no browsers natively support this feature, making prior solutions cumbersome and risky. To address these challenges, we propose $\texttt{RA-WEBs}$ ($\textbf{R}$emote $\textbf{A}$ttestation for $\textbf{Web}$ $\textbf{s}$ervices), a novel RA protocol designed for high compatibility with the current web ecosystem. $\texttt{RA-WEBs}$ leverages established web mechanisms for immediate deployability, enabling RA verification on existing browsers. We conduct a comprehensive security analysis, demonstrating $\texttt{RA-WEBs}$'s resilience against various threats. Our contributions include the $\texttt{RA-WEBs}$ proposal, a proof-of-concept implementation, an in-depth security analysis, and publicly available code for reproducible research.