Short explanation: That’s just another anti-Catholic lie. The New Testament was written by the Catholic Church, and it was the Catholic Church which decided which books are in the Bible and which books aren’t. The Church has always taught that the Bible is the divinely revealed word of God, intended for all people.
More complete explanation:
Even in the so-called “Dark Ages” when books had to be hand-copied and were thus terribly expensive, and when most people were illiterate, even then Catholic churches had a copy of the Holy Bible that was kept available for all to see. Like modern-day telephone directories in public phone booths, these Bibles were firmly attached with a chain so that nobody would walk off with them. This fact gave rise to the anti-Catholic lie that “the Catholic Church bound up the Bible with chains so that nobody could read it.” That’s idiotic. If the Church didn’t want anybody to read it, they would not have gone to such great expense to obtain a copy in the first place, nor to put it in public view! This shameless and silly lie about the Church is repeated so often that even many Catholics believe it.
If you research it, you’ll find that all the books and pamphlets which claim that the Catholic Church used to suppress the Bible either have no footnotes (e.g. the virulently anti-Catholic “Chick Tracts”) or they just reference each other, like the blind leading the blind. They never reference any “critical editions” of any historical works that are accepted by any mainstream historian (Catholic or not). Check it out; it’s true.
Q: If that’s true, then why don’t we use the Bible during Mass?
A: We do! On Sundays, we have a reading from the Old Testament, another reading from one of the four Gospels, and another reading from one of the other New Testament books. The “Responsorial Psalm” is from the book of Psalms in the Bible. The “Lord’s Prayer” is in the Bible. Many of the other prayers used during Mass are in the Bible too. Even the institution of the Eucharist is described in the Bible. We use the Bible a lot during Mass! Maybe we should point out that fact a little more often.