It is not an offence for a horse rider to leave the animal’s mess in the road. This applies whether the rider is an “over privileged middle class girl” apparently deliberately slowing the traffic (presumably because her horse cannot quite attain the national speed limit) or, as many riders are, a person with no particular advantages who has chosen to use some of their hard earned cash to pursue their hobby of horse riding.
Why it should be OK for horses to foul public areas but not OK for dogs to do so is a little mysterious (particularly in view of the relevant quantities involved). It could be that the laws to combat dog fouling are usually bye-laws introduced by local authorities to tackle a particular nuisance raised by residents. Additionally, although I am not an expert on horse sh1t (or any other kind for that matter, except the kind that some people suggest I spout from time to time), I believe DixieGal is correct when she says that dog's mess is far more toxic.
Where I live there are many stables round and about and a number of pavements beside dual carriageways which riders use to get to and from woods and heathland. There is often horse mess on these pavements but it is no big deal. It is easy to spot and soon washes away when it rains.
Lots of people round here collect the stuff in buckets and take it home. I believe they put it on their rhubarb, but I prefer custard on mine.