The other cultures don't want iron.
Iron rusts. Iron weapons break where bronze ones would bend. Iron's more expensive to manufacture and work than bronze.
If your culture has access to a good source of bronze, the downsides of iron might feel like they outweigh the upsides. Copper, tin, and arsenic are all easily smelted out of their ores (though your culture may not know these as separate metals, if the ores are usually found mixed up), and if you have a source of native bismuth then all the better.
Iron, meanwhile, needs higher temperatures than you can achieve in an open flame, meaning that you need to invest money into making high-tech furnaces--and then, you need to work it right or it gets really hard and brittle. If you put too much fuel in the furnace, or have too much air getting into the furnace, or have angered the moon god or what-have-you, you end up with iron that's either too soft or too brittle to be useful as a weapon.
They may hear stories of this big culture who use iron in all their weapons, but surely that can't be true--they'd never be as successful as they are if their swords shattered as soon as they hit someone's shield, and their breastplates fell apart as soon as it rained.
Depending on what the culture of the ironworking people is like, and how cunning their leaders can be, they may deliberately encourage this view of iron among other peoples, too. Here's a couple thoughts on ways they could do that:
All their soldiers also carry bronze weapons (either a full second spear or sword or what-have-you, or just a dagger for backup use) in addition to their normal iron ones. Someone who gets a glimpse of what seems to be a hidden bronze weapon during a military parade, or someone looting a battlefield coming across one of their fallen soldiers, might see that they carry bronze weapons as well, and feel like they've discovered some great secret, perhaps thinking the iron is just for show.
They could export some iron tools and weapons, but make them deliberately poor quality iron--even teach ironworking to others, but make sure they teach it wrongly, so that the iron comes out exceptionally brittle.
They could ensure that poor-quality iron (and deliberately poorly made weapons!) gets used in public displays, while the good stuff is reserved for use in actual warfare, so that the average person in their society, whose main experience with iron is seeing it used by gladiators (bronze is still the metal of choice for tools, of course), would develop an idea that iron weapons break frequently.