Safari not online although laptop is connected

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE 1: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

NOTE 2: if you have a wireless keyboard with rechargeable batteries connect it with its charging cable before booting into Safe Mode. This makes it act as a wired keyboard as will insure a successful boot into Safe Mode.

Go to System Settings->Network and check what appears there, particularly the DNS settings. Click Advanced, then DNS and tell us what you see.

Usually these DNS are either set to the router, showing just a local IP like 192.168.*.* or 10.*.*.* (which does not give away any private information, since these are reused locally by everyone), or to one of the known public DNS, like Google’s or OpenDNS’s, which are widely publicized so, again, that does not give away any private information.

You can take a look and just tell us in broad terms the format of the DNS if you don’t want to post the specifics.

On for Continuity purposes

Why can't Safari find Apple.com when I'm … - Apple Community