Being as this is what I do for a living, this thread caught my eye.
All the previous posts seem to summarize the obvious. And you seem to be aware of the importance of proper backfill compaction. 24" HDPE Type S storm pipe (like you have) is VERY sensitive to backfill and relies on it almost exculsively for rigidity. In summer heat, I am 100% certain the pipes will deform under the weight of your bridge. And I would be a little uncomfortable driving a heavy tractor across them. If the pipe fails, it will fail catastrophically (collapse into itself)under the weight of one of your wheels and given the 2 foot diameter, that would put one of your wheels quickly in a 2 foot deep hole. Pretty scary to me.
I am also quite certain that, as Mr. Walker indicated, this entire project will end up 'down the creek' as the pipes WILL simply float away (eventually) during a large storm.
Unfortunately, I don't see any solution other than to reconstruct your bridge, leaving sufficient gaps between the pipes for them to be "independently" backfilled, rather than as a bridge unit with the gaps unfilled (as they so clearly are now).
Without any upstream watershed analysis, capacity is unknown. Assuming you are OK with this general configuration, I would suggest digging down on your natural crossing, "seating" the invert (bottom) of the pipe into the ground by at least 6 inches. This will slightly diminish the capacity of the system, but it will eventually fill with sediment, helping to weight the very light pipes down. This also diminishes the possibility of water getting under the pipe. Most of your capacity is in the "gut" of the pipe, so if you're good now, you'll be good even if they're buried 6".
If you can armor the face with rip-rap, that would be best. Concrete works good, too. But the biggest things are filling the gaps between the pipes (a solid 1.5 or 2 feet of solidly packed soil between each pipe would be very helpful) as well as "seating" the pipes.