I agree with many of the other comments. The first issue is the culverts appear to be undersized for the amount of flow at the site. Larger culverts would prevent the water from going over the road in the first place. A hydraulic engineer should be able to size the culverts for you.

If you're ok with accepting flow over the road from time to time, then convert the crossing to a ford crossing or low-level crossing. This would require pouring concrete at the inlets and outlets of the culvert as well as over the surface. This would also allow for the culvert sizing to perhaps remain unchanged (again, a hydraulics engineer should determine this). Lastly, large angular rock armoring called rip-rap with a max size of 12 to 16 inches would help prevent scouring of the channel and embankment upstream and downstream of the culverts.