Piles of horse manure turning into compost appear to break down quickest when they are of cubic dimensions or nearly so.
My suggestion for turning horse manure into garden compost is to start with a base of some sort. If you do not have permanent or semi-permanent bins and need to have temporary sides, I suggest cardboard or many thicknesses of newspaper as the base, with 3 sturdy sides. The front of the bin could be made of boards etc. that enable its height to grow as you are shovelling in your quest for cubic “perfection”. 🙂
Piles of horse manure should always be covered (i) to conserve moisture (30 – 40% of the weight of the pile is water), and (ii) to prevent rain from entering the pile. If rodents are a problem, as they are for me, here, the cover might give you another layer of protection.