As mentioned suspend does work when I close the laptop screen (when I set the setting). Its only when I set the setting to turn off display, then the laptop’s display does not turn off at all when I close the screen.
If you go into /etc/systemd/logind.conf you will see a load of options. The one you want is: #HandleLidSwitch=suspend. Simply uncomment it (by removing the “#”) and change it to whatever option you want (probably “ignore” instead of “suspend”)