I had always assumed that since iron/”near steel” weapons were tougher and stayed sharper than bronze and an iron sword or axe can easily defeat the bronze weapon, armies so equipped won easily.
Iron was largely centered in Anatolia and would have changed the balance of power from that direction as soon as that metal’s forming techniques were mastered.
I saw a demonstration of iron vs bronze and not much of a difference - it is not like the iron weapon destroyed the bronze weapon and Greek armor was made of bronze well into Roman empire times.
The real reason the bronze age armies may have lost against iron equipped armies is that bronze was expensive and thus it limited how many troops a nation can have.
Iron being cheap could allow small nations to supply troops to create armies that could have as many if not more more troops than the established bronze empires. It was a case of mass production.