Since you don't own the azurewebsites.net domain (Microsoft owns it), you cannot make most the changes to map something.azurewebsites.net to another site you own.
However, you can run a reverse proxy app (such as ASP.NET Core/YARP) on something.azurewebsites.net to wrap over any other site you like. We can see from YARP GitHub issues that there are severa success stories.