PVC is brittle (more so in cold weather) and cannot flex much. I would never want to use it as a culvert pipe even if there was enough cover. The reinforced HDPE "dual wall" pipes are very strong and will flex instead of break.
4" pipe is a waste of time as it will plug up and then you'll have a real pain on your hands. I would not go smaller than 8" diameter, and 12-15" is the minimum allowed here.
If done right, the pipe won't be carrying traffic loads, only arch loads from supporting the fill/cover material.
If short on vertical space, you can dig the pipe down if needed -- don't overlook that little trick. A 12-15" pipe dug down 4-5" is still better than a smaller pipe.