The romaji for 大丈夫 is daijobu, but it sounds like daizhobu. Zh from chinese pronunciation, or like the “s” casual in english. Same for はじめまして - hajimemashite which sounds like hazhimemashite (i also don’t think that using the “zh” sound from chinese really makes sense here, but i don’t know what other letter sound like what in trying to explain )
It seems your concern is with how “j” is used in revised Hepburn romanization. Hepburn romanization is not systematic: it allows English speakers to approximate Japanese pronunciation without learning any new symbols, but that’s about all one can say in its favor. You might find Nihon shiki romanization more logical and easier to learn.
By the way, the linguistic term for that sound is a voiced alveolo-palatal fricative. In IPA, it is represented by ʑ, so you're closer than you think. :)
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