Change public DNS on EC2 Instance

I have an EC2 Ubuntu instance running a website. I also have a domain hosted with GoDaddy. I have setup a Elastic IP address which points to my instance. I have setup my domain in GoDaddy to point to the Elastic IP. This works and entering my domain takes me to the site host on amazon. But after the site loads the URL switches from:

www.mydomain.com

to the aws public DNS:

ec2XX-XXX-XXX-XXX.compute-1.amazonaws.com

I want the domain to stay as my domain and not switch to the aws public dns. I contacted GoDaddy support and they told me that this was happening on Amazons end so I must need to fix it somewhere in the aws dashboard. Can someone please give me the steps to get this setup or point me in the right direction.