how do I stop old earthlink email messages from coming into my new iphone 5s?

Earthlink is usually POP, therefore whatever you delete from a device or computer program does not sync up to your earthlink server. Log into you earthlink account on the web and delete and old emails you want gone forever. Then remove the earthlink account from your iPhone and re-add it back and only the emails left on the server will import to your iPhone. This is a limitation of email services that use POP and not IMAP, earthlink just never made the investment to upgrade their email services like the rest of the world.

If you don't use the Earthlink account, then just delete it. Go to Earthlink and delete your email account. If you still use the old account, then arrange to have your Earthlink new incoming mail forwarded to the email address you now use.