Conwy to Carnarfon by bus

I think you need Arriva Bus. I believe you're heading to Caernarfon, right? It appears that the A55 provides direct service. If it doesn't fit your schedule, a combination of 5C and 5 (or 5C and 5X) will apparently get you there, considerably more slowly, after a connection in Bangor. Be careful about the day of the week.

I took the bus from Caernarfon to Conwy a few months ago. It's a regular service taken by locals. It's confusing because the bus changes number in Bangor.

Services 5 and X5 connects with service 5C in Bangor for journeys to/from Caernarfon, through tickets are available, passengers do not need to change buses.

https://www.traveline.cymru is the best website for finding out how to do journeys in Wales. Also note that you could take the train from Conwy to Bangor and then a bus to Caernarfon. (This is likely to cost more than bus only. If using the train, hold out your hand at Conwy to tell the driver to stop. Inform guard on return that you wish to get off at Conwy. Virgin Trains do not stop at Conwy).