Third and fourth order rainbows need an improbable three and four reflections of sun rays inside the raindrops. At each internal reflection some light is lost, refracted outside of the drop to go towards a rainbow of lower order. The total light of the 3rd order bow is only 24% of the primary and that of the 4th order 15%. The higher order bows also get progressively wider (3rd order 4.4°, 4th 5.6°) spreading their light and weakening further the contrast between bow and sky.
Combine their low intensity with their position around the sun and it is no surprise that they were not photographed until 2011. Now they are imaged increasingly often. The 5th order bow has already been nailed to the trophy wall and surely even higher orders will be added.