Where is the data for your claims on accidents?  It's hard to imagine that head-on crashes of stage coaches were common occurrences. Accidents in competitions involving horses are fairly common, but bad outcomes on simple trips to town from the herders' camp to town for grub would most likely have involved bandits and other such hazards. And another thing, at the time when horses were in common use for transportation, traffic density was much lower than at the present time.  Riding through a deserted Fairmont Park in Philadelphia, a rider on a horse going at a walking pace and a rider in a car doing the speed limit would both be quite safe. Riders galloping would at most risk falling off going at about 25 mph. Drivers speeding with an equal degree of recklessness might spin out on a curve at 75 mpg. Riders can wear helmets that protect against brain damage but not broken necks. Drivers can use seat belts, and racing drivers can use other protections, so it is a little difficult to know what is being compared to what.  Even so, the deadly injuries among non-competition horse accidents are almost all head injuries suffered by riders whose horses spook for some reason. (See Jessica Jahiel's website.) The commonest injuries would most likely be broken arms from riders who fall.  A car crash at even relatively low speeds can produce the same head injuries (which is why cars now have padded dash boards), and as speeds increase the probability of serious injuries to the head and trunk becomes much higher.

As for feces, true, horse manure is not too appealing to walk through with bare feet, and somebody needs to remove it from the road. On the other hand, it is not toxic the way lead is. Once removed from the road, it can be used to produce good compost.  Even if people just leave it on a country road, it soon packs down into a uniform brown soft and well-compacted mass. Leave a clump of horse manure on your front yard and it become fertilizer. Dump a cup of used engine oil on your yard and you have long-term contamination.