Manure, garden compost and leaf mould are different and have different uses around the garden. Manure is high in nitrogen and other soil nutrients so is used where a high nutrient feed is needed, for fruit bushes, vegetables etc (except root veg of course). Garden compost has lower nutrient levels so is a gentler feed for seedlings and young pre-cropping plants, slower growing veg, and flowers and using as a mulch. Leaf mould is wonderful stuff - we haven't had any for 21 years

- and is mainly used for making your own seedling compost, for trays and pots. It can also be used for a low nutrient mulch to keep the surface loose and easy to hoe, as with a forest floor.
We have 3 bins in a row, the idea being to fill one, then when it's rotting well we turn it into the middle one (well, that's the idea) and start filling the original one again. Use the middle one then turn the first one into the now empty middle one and back to square one, or bin one. OR, once you have filled one, start to fill the one at the other end. When you need to start filling an empty bin again, turn the last one into the middle, then the first one on top of that - it will have shrunk enough to fit two bins-worth into one. That leaves you with two empty bins to fill while you use the well rotted lot.
It's generally easier to move the compost from one bin to its neighbour, but moving from one end to the other is possible.
For your five bins, you could keep two for manure. Horse dung doesn't take long to break down into a usable product. Once it's nearly there, turn it into the second bin and start filling the first one again. As your stable muck is arriving in bags, if you're not quite ready for it you can store it in the bags until you are. Or horse manure will stay in a free standing heap without a bin perfectly well - just cover it.
Leafmould really likes a more well aerated type of heap, hence usually being stored in a wire cage - choose the size to suit how much you have. leafmould takes up to three years to break down enough, especially if you've used slower decomposing species.
If you really want to compost brambles and twigs, then you're best to shred them and keep them separately, or at least put them in the bottom of an emptied bin before you start filling it. We prefer to burn woody plant waste and spread the ash under stone fruit and on onion and tomato beds, as a very valuable nutrient.
Each of your types of waste will rot down at different rates, so that's another reason to keep them apart.