The common solution is to include JavaScript code in client-side code that records the necessary information. This code can then capture metrics such as page load times, session data (the life cycle of an interaction that can span multiple web pages and operations), JavaScript and other client-side code exceptions, or AJAX time information.
The JavaScript code sends this data to a service that collects and retrieves and inspects this information.
Azure Application Insights follows this approach. All you have to do is to manually include calls to Application Insight API functions in the code of your Azure solution.
In order to make it as easy as possible to integrate the API calls, Microsoft offers you various Application Insight SDKs for various programming languages. All SDKs are open source projects and accessible via the following home page: https://github.com/Microsoft/ApplicationInsights-Home.