I get a little obsessive about clearing App Icon Badges on iOS and find them to be distracting, so I prefer to disable them.
How to disable an App Icon Badge for any iOS app:
Settings > Notifications > [App Name] > Badge App Icon > Disable
That's great but what about when your Twitter App Icon Badge App setting is missing? Bring back the setting by turning off Badging in the Twitter app:
Twitter > Me (Profile) > Settings (Gear Icon) > General > Badge App Icon > Turn Off