Assign Custom URL
Probably the single most asked question on the Help Forum relates to using custom domain URLs. It's quite straightforward provided you follow the correct instructions carefully. The instructions you need depend on whether your Site has been built using a 'free' Google account (a consumer account) or a Google Workspace account.
Consumer Accounts Users
Follow these instructions if you are using a free Google account:
https://support.google.com/sites/answer/9068867
or these instructions from Google Sites specialist, Steegle which includes a video
https://www.steegle.com/google-sites/how-to/assign-custom-url-domain-personal
OR
Google Workspace accounts users
Follow these instructions if you are using a Google Workspace account:
https://support.google.com/a/answer/7392423
or these instructions from Google Sites specialist, Steegle which includes a video
https://www.steegle.com/google-sites/how-to/map-domain-web-address-g-suite
Squarespace users, see these instructions
https://support.squarespace.com/hc/en-us/articles/21222002826381-Connecting-your-Squarespace-domain-to-a-Google-Sites-site
Godaddy users see https://www.steegle.com/google-sites/how-to/assign-custom-url-domain-personal/godaddy
Note 1: VERIFICATION: If you have not managed to verify your Site using the Webmaster Central verification method you are currently directed to by Google Sites, the above article uses a different method using the Search Console, which is easier/better. Use this method instead.
Note2: USE THE RIGHT ACCOUNT: Make sure you are signed in to the same account when verifying ownership of your domain that is the owner account of your Google Site
Note3: IT'S NOT INSTANTANEOUS: After adding the cname record it can take several hours for the change to propagate round the internet so your website may not be available on the new URL immediately.
Note4: CLOUDFLARE USERS: If you are using cloudflare use the 'DNS only' option, not 'proxied'
Note5: If you want to use the 'naked domain' (e.g. without using 'www') see 'Naked' domains