Cross-browser compatibility is the ability of a website or web application to function across different browsers and degrade gracefully when browser features are absent or lacking. When faced with cross-browser compatibility issues while developing a website using HTML,CSS and JavaScript we could follow a systematic approach to ensure a smooth user experience across different browsers.

Firstly, conduct thorough research on the target audience and their preferred browsers. This would help prioritize the browsers to focus on during the development process.

Next, adhere to web standards and best practices while writing HTML, CSS and JavaScript code. This would ensure that the website is compatible with most common web browsers.

To tackle specific compatibility issues, use feature detection techniques. By checking if a particular feature is supported by a browser before implementing it, provide alternate solutions or fallbacks for unsupported features.

Additionally, extensively test the website on different browsers and devices. This would involve using browser testing tools, such as BrowserStack or Sauce Labs, to stimulate various browser environments. By identifying and addressing any inconsistencies or bugs, ensure a consistent experience for users.

Furthermore, make use of CSS vendor prefixes and polyfills to handle browser-specific CSS and JavaScript website to work seamlessly across different browsers.

Lastly, stay updates with the latest browser updates and their compatibility changes. This would help proactively address any potential compatibility issues that may arise due to browser updates.

Overall, The approach is to involve a combination of research, adherence to standards, feature detection, extensive testing, and staying updates with browser changes to handle cross-browser compatibility issues effectively.