Azure Web Role is like a virtual private host. You can try getting a VM that acts as your web server, and you own that VM instance.
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In simple terms, we can say that Azure Web Sites are like an elastic shared hosting service. You can deploy your app to a web server that is not controlled by you and which also servers other users' sites.
You can also scale your site up and down (at some extra charge) to make it more elastic as your resource needs a shift.