If this email account is Hotmail:
Hotmail has used POP for emails in the past and this does not sync if messages are read or not. This is an old standard and shouldn't be used any more.
You can now use Exchange or IMAP for hotmail and this should mark them as read and stop this from happening again. Add a new mail account on your iPhone, and select Exchange instead of Hotmail/Outlook.com and sign in with your hotmail details. You may need to change the Mail Days to Sync options after you set it up to get all your emails back. Your hotmail account may also have put all your emails into a POP folder. Sign in to hotmail on a computer and move all the emails in that folder back into your inbox.
If exchange does not work, you can try using IMAP, the details are
To add the account, select other when you add a new mail account, and put in a nonsense email address for example "blah@blah.com" to avoid the mail app automatically setting up the account for you.
Then select IMAP and input the proper details. your username is your hotmail address.
If this email account is something other than Hotmail:
It sounds like your email account is using POP, try to find out if your email account supports IMAP and enter those details like i explain in the bottom part of the hotmail explanation.