Hi ChelsieSed
welcome to the Apple (user-to-user) Community!
Re: "Apple Watch logo flashes but can't restart"
You may want to check through this (non-Apple) article online (link below):
How to Fix a Flashing Apple Logo on an Apple Watch
https://macreports.com/how-to-fix-a-flashing-apple-logo-on-an-apple-watch/
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Re: .... it asked if I wanted to delete everything on the watch. There wasn't much on it so I did.
Now we're back to square one and the logo is flashing/blinking again."
Seems you will need to set up your watch again. You can also restore from a backup, if you wish.
You can set up your watch as new: Set up your Apple Watch - Apple Support
or, if needed:
See how to: Set up and pair your Apple Watch with iPhone - Apple Support
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More help or support if needed:
If you are near an Apple store you may want to make a reservation, and get help from the experts:
Genius Bar Reservation and Apple Support Options - Apple
or
You might want to reach out to the Apple Watch Support Team (online chat / phoine / email etc.)
There is a "Get Support" link (scroll to the very bottom) on their webpage (link below)
Apple Watch - Official Apple Support
All the best :-)