I would expect the word order here to be: 'IHmo' Qat turghal. I don't see how the person can be the object of the verb here.
He's not; he's the subject.
'IHmo' turghal, Qat.
I'm not so fond of your example because when you start hearing 'IHmo'... you don't know who we're talking about. 'IHmo', Qat turghal -- is Turgal popular because someone else ("she" or "he" or "they") is/are handsome? Or because "it" is beautiful (perhaps the building he designed?)?
Okay, I can see what you're going for here, but that's now how the English sentence reads. The focus of the sentence is "Turgal is popular" which requires Qat turghal with "because he is handsome" as the clause.
If Klingon had punctuation, then a comma (which you've added in your explanation here) might make a case for parsing it as you have, but that still leaves the misleading English sentence.
If it's considered important, the solution is obvious: put the subordinate clause at the end. Qat turghal 'IHmo'. Now who is beautiful is exactly as unclear as in the English. In both languages, context will inform your understanding.
As mizinamo points out, it's important to remember that the reversed basic sentence order of Klingon means you may have to rearrange some words to keep a pronoun's antecedent actually ante. Sure, postcedents exist, but they're usually an irritation, and TKD makes a point of saying that you typically turn subsequent invocations of a noun into a pronoun, not the first one.
But this is more a matter of style than absolute correctness.
It really seems like the English representation is what is mismatched and causing the confusion. "Because Turgal is beautiful he is popular." Now the word order in Klingon makes perfect sense.