You probably could, but the amount of engineering and cost required would be simpler to just replace the bridge with a stronger one.

Often these "car only" country bridges are just some large wooden beams over a creek; even a simple concrete culvert replacement would be truck-proof.

Some of them do things like this: https://www.alamy.com/stock-photo-small-car-suv-crossing-the...

A really simple catch all, that isn't quite accurate for all cases would be to put a sign over the top of such a bridge,that limits the height of vehicles crossing, excluding trucks with the tall trailers. Would not work for loads that aren't tall though.

I would have thought a few strain gauges, maybe some logic to filter false positives (if that's a risk) and a light would do a reasonable job.