If you are hosting unrelated websites for multiple people - i.e. you are the server owner, and offer webhosting for customers or friends - putting each user's website under their home directory makes some sense, as you can then e.g. easily use the SFTP chroot jails.
However, that makes sharing access to website code between different usernames very problematic.
If your webserver has a single, primary goal, or if you need to give multiple users access to a website, it's much better to user /srv (if such directory does not exist, feel free to create it).
If you use /srv, your websites are not tied to users; however, when you're using sftp chroot, you will need to bind-mount the website directory to each user's home directory so they can reach to it from inside of the chroot.
Also, f you're granting multiple users access to website code, you may end up with some trouble when different users create files, and put different permissions on them - this can be mostly mitigated by setting up appropriate ACLs on the website directories.
For a concrete, if a bit complicated example, one of my webservers is a mix that hosts a bunch of "main" sites and some separate ones for friends.
Main ones are in /srv, and are being edited by our core team (multiple users); we have ACLs in place to minimize the permission issues, shell access, and non-chrooted sftp.
Friends have theirs inside their homedirs, and are chrooted so can't access anything outside of their homedir.
As of moving the existing website, well, depends on what the website is.
Some CMS systems are known to hardcode disk path in various places in the database, for example.
But in general - create /srv/websitename , move files from current location to there, adjust the webserver's configuration to use that directory, and see if it works.

If it doesn't, time to investigate.
Are the permissions on the files set so webserver can read all it needs to read, and write where it needs to write? (Be very careful about the second part, write access should be as limited as possible).
Can you see some errors referring to the old path? Then there are some references to old path in some config files of the webserver or website (or in the DB). This has to be edited.
If you're using some popular package, like Wordpress, or Drupal, or such, asking for help in their forums would give you more concrete information.