How can we believe and trust censored Bibles?

Hehe

Yes, of course.

Both of these things need defining before anybody can answer your question.

"Censoring", the way I understand the word, means that there's some kind of institution charged with overseeing and removing parts of a text. So I wonder at which point in the development of the Bible you believe this has occurred.

I've argued in a different comment that it's no secret that certain texts were picked and chosen by the early church as part of its canon, but that (in my opinion) is a very different thing than censoring. To give an analogy: If I was an editor and had to choose the "100 greatest novels of the 20th century" for a book, I would not be "censoring" those I didn't choose. Therefore I'm asking you what exactly you mean by censoring, and if you can give examples of censorship happening in the development of the Biblical texts.

Secondly, "original Bible" is not at all easy to define. The (Christian) Bible is a collection of texts of diverse genres, by a multitude of authors, in three languages, spanning at least seven or eight centuries in their development. None of the original manuscripts have survived. Instead, for every part of the Bible, there exist different copies which sometimes differ slightly, sometimes starkly. This is the reason textual criticism of the Bible exists as a field of scholarship. Most notably, the (older) Septuagint version of the Book of Jeremiah is about one eighth shorter than the (later) version of the Masoretic text.

All of this means that if you're going to talk about the "original Bible", you have to tell us what you mean by that. Do you mean

• the original manuscripts of each individual book or passage, all of which are lost?
• the oldest surviving copies of each passage, respectively?
• the Septuagint (and if yes, which version of it)?
• the Masoretic text (and if yes, which version of it)?
• the current scholarly consensus on the most faithful manuscripts, as collected e. g. in the Biblia Hebraica Stuttgartensia and the Novum Testamentum Graece 28?

King James notoriously removed mentions of the word tyrant in his English translations.

AFAIK this is an urban myth. But even if true, it's hardly a case of "censoring", but more a (questionable) translation choice. (Because "tyrant" is not a word that appears in the original Hebrew or Greek, so it can't have been censored in that sense.)