The roman empire ( eastern portion anyways ) lasted until the ottoman turks took them out in the 1400s. So the crusades definitely happened during the roman empire's existence.

I think Roman Empire was split clearly East/West.

Crusade came from the West. Roman Empire, or what the last vestiges of, fizzled in the East.

Not acknowledging the Roman Empire when it was seated in Constantinople is kind of peak Western chauvinism though.