Yes, twit is a Scrabble word!
...and is worth 7 points in Scrabble, and 7 points in Words with Friends
twit
Verb
1. Harass with persistent criticism or carping
Noun
1. Someone who is regarded as contemptible
2. Aggravation by deriding or mocking or criticizing
What Scrabble words can I make with the letters in twit?
3 Letter Words
2 Letter Words