Although it may come as a big surprise to many American evangelicals, when the Pilgrims came to America in 1620, they did not bring with them the King James Version of the Bible. Instead, they brought with them the old Geneva Bible.
The Pilgrims, being Puritans, were sympathizers with the Protestant Reformers. Therefore, they still preferred the Geneva Bible, which was translated by the Reformers. They had serious doubts about the King James Version, since it was the new authorized version of the Church of England, a church that not only opposed the Protestant Reformation, but had severely persecuted them, the Puritans. In fact, it was their flight from such persecution that led them to sail the sea in search of a new world where they could find religious liberty.