How to make an Azure hosted ASP.NET website use my domain name as its base URL?

I have a website hosted on Azure that is written on ASP.NET

I have registered a domain name and got that domain to redirect to my azure hosted website

How do I get the website to use my domain for its base of its URLs?

(For example how would I make the about page be http://www.taskbazaar.ie/Home/About - not http://taskbazaar.azurewebsites.net/Home/About ?)

Do you have access to running INETMGR / IIS with an AZURE account?

No - but there's a configuration screen on the Azure portal that can set many such settings (things like connection strings etc.)
However from reading around the subject it appears that this is not possible for the free website hosting - you have to be on at least "standard" to set your own domain.

I just went down this same exact path with GoDaddy. Tried the simple SHARED server with IIS and could not configure it nor install my particular dependencies.

That had a web portal to manage things like connection strings.

I had to switch to a GoDaddy virtual server - where I get actual REMOTE DESKTOP into a WINDOWS 2008 SERVER.

Been a nice experience so far - the RDP can be a bit laggy - not sure why - but the website responds nicely.

You could create a static html on the hoster where you've registered the domain. In that page you create an iframe where the source is the url of the azure site.