BBC: what syllable do 'shed', 'she', 'fishes' and 'masher' have in common?

Linguistics is almost invisible in the media and ordinary discourse. Popular discussion of language treats it as a collection of words (made out of letters) which are used to make sentences (a process which apparently should be governed by rules but often isn’t).
Even when there’s a good article, false folk notions tend to surface. For example, this fairly good discussion of how to pronounce Chinese politicians’ names (which gives mostly decent pronunciation advice*, and explains tones succinctly and accurately) fails badly at the end:

Hanyu Pinyin's relationship between spelling and pronunciation is not as random as it seems because it is actually syllable-based; the syllable xi is always pronounced shee (-sh as in ship, -ee as in meet), whether in Xí Jìnpíng or móxī (the Chinese name for Moses).

OK so far, but…

Compared with the wide variation one finds for the English syllable she in the words shed, she, fishes and masher, pronouncing Pinyin can suddenly seem much more pleasingly systematic and straightforward. 

Do I need to point out that the string of letters ‘she’ is not a syllable, and that the examples chosen, ironically, make that point? I suppose the problem here is collision of the folk notion that words are made from letters with the linguistic notion that they can be decomposed into syllables.

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* But the vowel in the Mandarin rendition of ‘London’, glossed reasonably enough as ‘luun duun’, is not the one from ‘book’. It’s closer to the one from ‘her’ (in non-rhotic accents of English) with an approximant at the beginning. Something like: Lwerndwern