I recently set up an Azure web app as a splash page for a new product I'm launching. I wanted to use my custom domain but everything I was finding online was out of date with how to set it up in Namecheap, my DNS provider of choice. I finally chatted with Namecheap support who had me up and going in a few minutes. Here's what you do:
First, open the Azure management portal, navigate to your web app, and click on "Configure" at the top. Scroll down until you see the "domain names" section and click the "manage domains" button as shown here:
Copy the ip address out of that box, as highlighted here:
Next, go to Namecheap and click "Manage" next to the domain you want to use. Click "Advanced DNS" and add the following entries:
For the A record, you'll paste in the IP address you copied at the beginning. Be sure to save all those entries and then give it 24 hours.
If you want to check DNS propagation, you can use digwebinterface.com to do so.
Additionally, here is the official Azure post about how to do this in general.