Can anybody explain how this sentence translation is "An Arab American woman" and not "An American Arab woman"?
('bint' is 'girl' rather than 'woman')
That said, this is simply a matter of adjective order which differs in usage between languages (like in English it is quantity-quality/value-size-temperature-age-shape-color-Proper (nationality/etc.)-material-purpose. You would say I have 'few charming old round Victorian silvers' not I have 'Victorian round charming old few silvers'). It is just a matter of traditional usage.
'Arab American people' as a phrase in English is understood generally to mean Americans of Arab origins (without getting much into identity definitions), where they themselves choose to identify themselves in this word order.
The equivalent of that, in meaning, in Arabic is 'American-Arab people'
So the transliteration of this sentence is 'an American Arab girl' The translation of meaning is 'an Arab American girl'