Deleting email messages

Are you dealing with just one email account or more than one? If more than one, are you seeing the same thing on all email accounts?

Is it possible your email account is set up as a POP email account, rather than IMAP? A POP email account loads the email messages onto your Mac where they're stored and operated on locally. An IMAP account stores the emails on your ISP's server, and can therefore be deleted by any device, where it will be deleted on all devices.

If both are signed into the same AppleID, deleting them from one should delete them from the other, too.