Safari keeps quiting

Are you running any anti - virus programs? Using a VPN?

Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account 

Safe Mode - About

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin

Etrecheck – System Information

You can try Etrecheck, but since it is crashing in another admin account and the Safe Mode, you may need to reinstall the OS. Do a backup, preferable 2 separate ones on 2 drives. Boot to the Recovery Volume (command - R on a restart). Run Disk Utility and select First Aid. Then re-install the OS.

Reinstall from macOS Recovery

OS X Recovery

I have the same issue. One day starting up Safari this week it just wouldn't. Crashed immediately. Tired safe mode and no go. Tried a new Admin account and safe mode and still no go. AdBlock is the only extension I have. I'm not familiar with a way to shut it off without accessing the program. Would the Etrecheck testing work in my case? Thank you for any help!