Bronze and iron weren't limited for military usage. Basically from an economical point of view, the scarcity of bronze limited the size of economies and armies.

Once people started using iron instead of bronze, the first use was probably creating tools used for farming, food production, and other important things: Axes, hoes, spades, etc. And they did not just substitute bronze tools but because iron was cheaper, they probably created more of them, and new tools. And it also enabled things like nails and other construction material.

So, the economy and power structures shifted as tools enabled more economic output. By the time steel became good enough for weaponry, the bronze age was effectively already a thing of the past as the economy would have shifted to using iron for most tools long before.

And then later of course, having cheap and good enough steel weaponry enabled the creation of larger armies too. So towards the end of the Iron age, you get e.g. the Romans creating huge legions that were simply not economically feasible in the bronze age. And given that hand to hand combat was basically the way people waged wars, the ones with the bigger armies would tend to win.