Delete all your mail. Then do a hard reset. Hold the top power/sleep button and the home button at the same time until the Apple logo appears, then release. Now start over with your mail. If it continues to show read, then 1. Someone is accessing your mail, or there's an issue with the account. Do you have other email accounts? Do they do the same thing? I have one of my Wife's email accounts on my phone. Verizon had me add it on to check an issue. I never bothered to delete it. She doesn't use email much. Mostly spam ads. But you see how someone else may have added your account to their device. A spouse? 2. It may be nothing. Try the hard reset.
Sent from my ancient but trustworthy iPhone 5.