I believe it is important, for all riders who must ride on public roads, to look up the laws in their particular state. Each state laws are slightly different. A few have no laws at all, regarding horses. Some have laws which differ from riding a horse, to one who is driving a horse-drawn vehicle. In some states, a rider is required to ride 'with' the traffic. Others require the rider rides against the traffic.
But regardless of the laws in any given state or even country, every rider should be very aware, that the majority of motorists one will meet, are not horsey. They might even get a thrill from trying to scare the horse and rider. In most areas, motorists are required to slow down and give a horse a wide berth when passing. Few know that law or bother to do so.
I suggest anyone who must ride on roads, first try as often as possible, to ride inside a fenced area, along a roadside, so their horses become very used to all kinds of traffic, the noise and speed. It takes time and patience, to train a 'traffic safe' horse and even if solid, one still has to deal with the idiots who will try to spook a horse.
Sometimes, if you live in an area where many horses are constantly ridden on public roads, you can encourage your local officials, to post signs to warn of horses. After all, many areas have such signs for deer.