Shi

This is not pronounced like the English word "she", but more like "shirt" without the "t" and without moving your jaw. IPA: [ʂɻ̩] or [ʂɨ].

I agree with the user raincoatdance that the pronunciation currently listed is wrong. The consonant is right but the vowel is wrong. The tongue is further back in the mouth in "shi" than in "ee" ('i' in IPA), and I agree that it sounds more like "sher" in English but with your mouth more closed.
As for the IPA, it seems like there's a few different ways to represent this sound (they call it an "apical vowel") in Standard Mandarin via IPA. I think the most widely agreed upon is /ʂɻ̩/, but also used I've also seen /ʂʅ/, /ʂʐ̩/, and /ʂɨ/ used.
Here's some sources I used for the IPA stuff:
https://en.m.wikipedia.org/wiki/Help:IPA/Mandarin
https://en.m.wikipedia.org/wiki/Standard_Chinese_phonology#Syllabic_consonants (cites specific instances when different IPA is used in different academic works)
https://resources.allsetlearning.com/chinese/pronunciation/The_%22ch%22_%22sh%22_and_%22zh%22_sounds
https://www.reddit.com/r/linguistics/comments/1mzuoj/the_i_in_mandarin_wikipedias_ipa_page_seems_wrong/?rdt=57902 (reddit post where people discuss the different IPA representations of the apical vowel) [noted -ed]