This definitely isn't normal behavior for the Mail app. First, try closing the Mail app by double-clicking the home button and swiping up on the Mail application window. If the app still behaves strangely after reopening, then we can try to hard reset your iPhone (this won't make you lose any data). Press and hold the home button and power button together and release once the Apple logo appears on your device.
If this doesn't work, I would suggest going into Settings>Mail, Contacts, Calendars then under "Accounts" tap on each mail account and toggle off and on "Mail." I hope these steps will help solve your problem.