According to wiktionary: http://en.wiktionary.org/wiki/ya%C5%9Fl%C4%B1
yaşlı came from yaş + -lı.
And yaş means year: 1.age (part of the duration of a being or thing between its beginning and any given time)
In Turkish, it seems to me that the "age" of a being is a character you carry around rather than an ephemeral state you're in. I'm not a linguist, but this is similar to 阶段体(phase aspect?) in modern Chinese, which describes the duration of the verb.
I suddenly remembered the latin "natus esse" construction, e.g. annos undeviginti natus (.RES GESTAE DIVI AVGVSTI), which translates to "In the time of having been born for nineteen years". But English won't accept the construction "having been born", so... I guess this is a small difference.
And the French counterpart is J'ai dix-huit ans (I have eighteen years).