No, according to the
SearchTrek transcript-searching app, "turbolift" was used frequently throughout all the Berman-era shows,
Discovery, Picard, and
Lower Decks, as well as the latter two Kelvin movies, although it doesn't include transcripts for any of the other Secret Hideout shows. Surprisingly, it was only used twice in dialogue in TOS (in "The Naked Time" and "By Any Other Name," plus one use of "turbo-elevators" in "Space Seed"), and never in TAS. (The above search link doesn't include the "Naked Time" instance because the transcript has it written as two words. I found it by searching for "turbo.")
Which makes it odd that it caught on as the default term, given that
The Making of Star Trek (which was generally the source of terms that became standard despite not being used much in TOS, such as "mind-meld") called them "turbo-elevators." But
The Star Trek Concordance used "turbolift" in its lexicon, so that's probably why it caught on as standard.