If you absolutely need your website to be in /home/$user/...
, then you could change the permission of your directory like this :
Create a group for this directory
groupadd home_user
add your user, and the user executing the CMS (e.g. wordpress) inside this group.
usermod -A -g $user home_user
usermod -A -g wordpress home_user
Then change the permission of your home folder:
chown $USER:home_user /home/$USER
chmod 770 /home/$USER # or 750 if it is enough
Then it should work.
BUT !
Having your website in the home folder, this is bad practice.
It would be better to have it under /var/www/
or /www/
or /opt/www/
Then it is easier to segregate. You give the authorisation to this folder ONLY to the user which is running the website. And he has no other permission.