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Twitter’s archives are now fully archived and searchable. Anyone can search for tweets back to 2006, when the service began, at https://twitter.com/search-home.
The Advanced Search page offers some interesting options, including place specification, date limiters, and positive vs. negative tone: https://twitter.com/search-advanced.
Rest assured that you can remove tweets from the public collection (although do remember that the Library of Congress maintains an archival, non-editable, and currently non-searchable collection of tweets). Just sign into your Twitter account and locate any questionable tweets from your past. Then delete them. This could be quite a chore for some of us.