It was revealed that people were seeing their toots in Google search by simply googling their username & Switter together.
If you don't want your toots out in the open, then here's short guide!
Go to your profile and look for the 3 lines. Click on them, and hit Edit Profile.
View of the Edit Profile page. Look at the navigation on the left.
Click on Preferences underneath Edit Profile.
View of the Preferences page. Scroll down to the Other section.
Checkmark Opt-out of search engine indexing to keep your profile & toots private!
After checking the box, click Save Changes at the bottom!