Do you have an iMAP account like a Gmail, Yahoo, Outlook? If so then try this:
Go to Settings > Mail, Contacts, Calendars > locate your email provider and click on the account. Then on the next page Delete Account.
Then do a forced restart. Hold down the Home and Wake/Sleep buttons at the same time for about 15-20 seconds until the Apple logo appears. Ignore the "Slide to power off" text if it comes up. You will not lose anything.
Then go back to Settings > Mail… and add the account again.
Then go to Settings > Safari > Clear History and Website Data.
And now check and see if you now can open the links directly.
If this does not work, you may have to Restore your phone by following the directions below. It can look daunting but iTunes walks you through it pretty easily. However it will take about 30-60 minutes so let's hope the steps above take care of it.
Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support