I didn't realize I needed to explain more, jeez.
I was wrong, it was actually turn 16 or 17 where I took Caernarvon. I had taken York already, as well as Rennes down in the mainland. I know I could've forced a sally, but I was feeling impatient and they still had several years worth of supplies.
I did indeed have longbowmen and billmen. The longbowmen were good (and even better in my newest battle, I'll detail in a second) and the billmen did their part but were pretty torn up by the end of the siege.
The newest battle, I've declared war on Scotland. I started it off by attacking a Scottish fleet with a huge army on-board, hoping to wipe out the bulk of their forces before the real fighting began. It didn't work, no ships were lost on either side and we had to withdraw. That huge army is still floating around somewhere, I lost line of sight on them and I'm kind of worried about that. Thankfully it's a Scottish army, so it's mostly rabble.
I decided to hit Edinburgh with my new king (William died right as the war started), Rufus, while the main force was playing sailor. The enemy garrison was four cavalry units led by a family member, so I didn't see how they'd stand a chance against my new army (Much more infantry based than before, lots of spears and billhooks, plus longbowmen) To my great joy, I love when they do this, a second Scottish army attacked (three cavalry), forcing the enemy to sally forth after only a turn of sieging. Since they're still the garrison, if they lose (And they will) they'll all die and I'll take the city. In my old game as the French some Italians (Milan or Venice, I don't remember which) killed their faction leader like two or three times doing this. They just never learn.
The battle is a massacre so far, I set up stakes with the longbowmen and put my spearmen on the flanks, the three cavalry units lasted all of thirty seconds before routing with less than half of their forces, and the family member is on his way, leaving me plenty of time to redeploy and wipe him out too. I'll have to leave the stakes behind though, since he's coming in from a weird angle.
Aaaaand Rufus killed himself on those stakes. Seriously. Somehow when they were charging he managed to walk right into them. It's my fault actually, I didn't bother moving my cavalry as I didn't think I'd need them. Dumb.
Aaaand my man of the hour, Captain Edmund, was so excited by the news of his adoption into the royal family that he died. Great. That battle did not turn out as well as it started.