All that changed in 1878 when photographer Eadweard Muybridge (who had a strange addiction to vowels) was called in to settle a debate on what exactly horses did with their feet while trotting and galloping, and how many feet they had off the ground at any given time. Muybridge, a pioneer in motion pictures, took high-speed photographs of a horse running. The result was a beautiful and extremely famous frame-by-frame depiction of the gallop.
Note that the only time all four feet are off the ground is when they are all under the horse. The rest of the time, at least one foot is touching down. The above depicted fully extended portion of the gate did not exist. This was rather widely publicized, and from then on horses in art were depicted running properly. This is a John Sanderson-Wells painting from sometime after 1895:
When I go to antique stores, I like spotting the pre-1878 flying horse paintings.