The documents show Cardinal Roger Mahony's frustration as he tried to ensure one priest, the Rev. Kevin Barmasse, did not return to pastoral work.
Barmasse, who allegedly abused at least eight teenage boys after plying them with alcohol, appealed to Rome.
"Given the pastoral situation in the United States today, which is all too well known, bishops need to be able to act quickly and decisively in cases of alleged clerical misconduct to assure the People of God that their rights are being fully protected," Mahony wrote in a 1994 letter to a Vatican official the cardinal had met with during a visit to Rome four months earlier.
Under Catholic canon law, bishops can remove priests from parishes but only the pope may remove them from the priesthood. Barmasse was eventually unfrocked, after more than a decade.
John Allen, a correspondent for the National Catholic Reporter, told the Times the delay was not unusual.
"This was not just Mahony's experience," Allen said. "Anyone in the world who had dealings with the Vatican in the '80s and '90s was frustrated -- who's in charge, what's the procedure, how long it took."
Bishops had to deal with multiple agencies, including the Congregation for the Doctrine of the Faith, headed at the time by Cardinal Josef Ratzinger, the future Pope Benedict XVI. Mahony retired in 2011.