Bitton was born in, and grew up in the area of, Blackfoot, Idaho. He was a talented pianist, having begun at the age of six. After two years at Brigham Young University (BYU), he served as an LDS missionary to France, where he edited the church's L'Etoile periodical. He then served in the United States Army during the Korean War.Bitton returned to BYU, where he was president of his Phi Alpha Theta chapter and graduated in history in 1956. He afterward studied at Princeton University, where he received an M.A. in 1958 and earned his Ph.D. in French History in 1961.

Bitton was a professor of history at the University of Texas at Austin until 1964, when he moved joined the University of California, Santa Barbara. He then joined the University of Utah faculty in 1966, where he taught for 29 years until his retirement in 1995. From 2005—2006, he was a visiting professor at Brigham Young University Hawaii.

Although his specialty was French history, Bitton made many contributions to Mormon history. He won many awards for his work in Mormon history, including the "Silver Award" from A Journal of Mormon Thought, awarded for an essay on B. H. Roberts. He was an original member and founder of the Mormon History Association in 1965 and he served as president from 1971—1972.

Bitton served as an official Assistant Church Historian to his mentor Leonard J. Arrington from 1972—1982. Bitton referred to this time as "Camelot", an exciting time of unprecedented development of new Mormon historical research. During this period and after, Bitton published several works with Arrington. In 2006, the Mormon History Association awarded Bitton the Leonard J. Arrington Award for "for distinguished and meritorious service to Mormon history".

Bitton married his wife JoAn in 1984, and later in life they served together as guides on Temple Square for five years.

He died at the age of 77 in Salt Lake City.

Books

• Winner of Outstanding Bibliography Award (Mormon History Association)

• Winner of Best Book Award (Mormon History Association)

• Winner of Best Book Award (Mormon History Association) and Evans Biography Award (Utah State University)

Articles

• Winner of Best Article by a Senior Author (Mormon History Association)