In general, the opposite of what you say is true: the more occupied bandwidth, the less power you need for equivalent information transfer. So, e.g. when the FCC made everyone doing VHF/UHF voice comms narrowband from 25KHz to 12.5KHz, you needed basically double the power for equivalent analog voice.

Or for an extreme-- GPS uses 50MHz wide to send dozens of bits per second to allow much lower satellite output power and resilience to noise.