It’s possible to take this too far, like mispronouncing “Beijing” as “bei-zhing” because it sounds more foreign and gives the impression of being educated and well-traveled.
This will sound weird, but if I say it "in English" it's Bei-zhing, and if I say it in Chinese it's Běi-jīng, and that J isn't really a phoneme we have in English anyway. So nobody's really pronouncing it right.
A name from a tonal language will never sound entirely right in English, though I got used to pronouncing “-jing” with the “j” sound from “jiffy”. For a while, I thought the “zh” sound was correct, partly because other people said it that way, and partly because it sounded plausibly exotic, until I discovered that there was actually a phenomenon for this mistake called hyperforeignism.
I'm familiar with the phenomenon, what I'm saying is "j" in pinyin represents a phoneme that we don't actually use in English. Ironically, the "zh" sound in pinyin is probably a closer approximation to the "j" sound in "jiffy". The hyperforeignism of Beijing seems to be from misapplying the reading rules for French, not Mandarin, so it's an interesting case, but both pronunciations are only approximations.