First Bus confirmed the vehicle involved was the company’s X11 service, which runs between Carmarthen and Swansea.
Police confirmed all passengers on board the bus involved had been “safely removed” and were receiving “appropriate care”.
A rest centre has been set up at the nearby John Burns Centre.
Mid and West Wales Fire and Rescue Service said it had sent three crews, from Tumble, Carmarthen and Kidwelly fire stations.
Hywel Dda health board said its teams were responding to the incident.
“Our emergency departments are busy, and waiting times are likely to be longer as we treat individuals with the greatest need first,” it said.
Local vicar Father Jim Flanagan said he was re-routed due to the road closures and saw the bus over a hedge.
He added there were multiple ambulances there, with lights flashing, and an air ambulance had landed in a field.
One of his churches, St Mary’s in Kidwelly, has been opened should people wish to use it, he said.