Continuous security alerts--are they real?

No, they're scam messages.

If these are sliding in the from the top right of the screen, a site you visited has given itself permission to send push notifications. Everything you're seeing is nothing but a scam.

Open Safari's preferences (Settings). Click on the Websites tab and then scroll down to Notifications. Clear any entries in the right hand window. You might find something similar in the Pop-up Windows heading below that. Block any you don't recognize.

Then if you want, uncheck the box to the right, as shown below here.

I personally can't think of any reason why I would ever want any website to push notices to me.