Adding a Custom Domain in Google Sites 

Showing the process needed to add your custom domain to your Google Sites. I never find it the easiest process because you have to be patient and wait for it to go through.
The 3 things you need to do is:
1. Go to settings in your Google Sites and select Custom Domain
2. Go to Google Search Console and verify ownership of the domain.
3. Go to your URL provider (GoDaddy, Squarespace, etc) and edit the DNS settings and put in the information that Google Search Console provided using the CNAME method.