Merriam-Webster pronounces "transient" as \ˈtran(t)-sh(ē-)ənt\.
However, most Americans pronounce it as \ˈtran-zē-ənt\.
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Merriam-Webster seems a bit of an outlier with its intrusive t, though Cambridge UK adds a d in the recorded audio of its suggested American pronunciation. It sounds like tangent only with an intial tr. The IPA for that pronunciation is given as
which, I trust, is a typographical error. Or a really drunk guy trying for transigent.
On their American page, there are three choices, one with the added dental consonant:
ˈtræn·ʃənt, -dʒənt, -zi·ənt
So we have two authorities who add a d or t.
The American pronunciation depends on whether one gives the word three syllables or, in more rapid speech, just two, in which case the i disappears, transforming the s into a palato-alveolar fricative, either voiced or unvoiced.
The American three-syllable pronunciation voices the s just as the British:
ˈtræn zi ənt
There are a number of examples at FreeDictionary.com, which aggregates several online dictionaries. When reduced to two syllables, the s palatalizes to either ʃ or ʒ by a process called yod-coalescence a fairly common feature in English: vision, measure, education, etc. This yields:
Now the pronunciations with the added dental stop both sound odd to me, but I assume they’re used somewhere in the US or Canada. Otherwise these dictionaries wouldn’t bother to list them.