Because dum iúvenes súmus does not fit the metre. The verse is stress-based, and iuvenes is stressed on the first syllable.
Not sure what to make of the objection that the same word order “sounds very ungrammatical” in English or Croatian. That would be surely true of many or even most Latin sentences. It's almost as if the ancient Romans didn't consider English grammar when they invented Latin. It is no criterion for reasoning about Latin syntax.
In Latin, anyway, it's entirely allowed that a conjunction is not the first word of the (subordinate) sentence. It's quite common for relative or demonstrative pronouns to be put in front of it, but occasionally it's other words or even whole phrases, e.g. Ad hoc consilium cum plerique accederent, Histiaeus Milesius [⋯] obstitit (Nepos, Miltiades 3).