A riding horse on its own is considered Large in game terms (20 square feet). When it's drawing a cart, what size is it considered to be?
Carts are Large objects, per the table of sample object sizes on p. 247 of the DMG.
So a horse with a cart is a Large creature pulling a Large object, occupying a 10' wide x 20' long area.
Note that this isn't treated as a single creature, since creatures can squeeze to move through spaces one size smaller than themselves. Objects, including carts, cannot squeeze.
There are no specific rules for how a horse and cart can move (in terms of actions, speed, or limits to turning). D&D 5e generally prefers to leave details like that up to the DM.