It's already ha-ji-me-ma-shi-te. There is not only "j" in Japanese. The "i" is not that audible, that's all. When you say "hajimemashite" to yourself a few times you'll see what I mean. Plus, sometimes romaji texts may not have the correct spelling, because they're written for English speakers only. For example 大橋 is おうはし (ouhashi) but on a sign I saw it was written "ohashi" in romaji, supposing a foreigner who may need to see the romaji would already pronounce it as "oh-hashi". The difference is, "ouhashi" means "the big bridge" but お橋 (o-hashi) would be "the bridge" only with a... prefix showing respect. So, don't pay much attention to the romaji. There are even Japanese people who can't spell words in romaji, and it's not a fault.