The journey to the abandoned royal palace was to be by horse. They did have a carriage available if needed, but according to Jhen, the terrain wasn't suited for them.
"We'll have to travel through rough terrain to get there, so a carriage will only slow us down. It's best to go just with the horses."
William didn't mind. He actually preferred traveling by horse. Inside a carriage, he felt like he was missing out on the beautiful scenery, just sitting there doing nothing. It was quite boring for him.
At the same time, traveling that way meant they'd have to stop more often to rest. It was tiring to ride a horse for hours non-stop.