As far as I know, the best and simplest priority splitter is a single wire after your splitter.
Video linked above uses this approach, though with so much useless stuff on the screen it might be hard to understand how simple the basic idea is.
You use a single wire (red or green) to connect both belt tiles after the splitter. One of them will be prioritized, the other won't.
You need prioritized belt to always run, so you click on it, remove blocking operation mode and set it to read contents in holding mode (so the signal is generated continuously).
Other belt is left in enable/disable mode, it must NOT read belt contents (so the wire only counts items on the prioritized belt tile). The condition for enabling is that resource that you're carrying on the belt is greater than N, where N will usually be within 2-6 range depending on whether your prioritized belt is straight or corner, whether the resource occupies both lanes or only one lane and which one specifically. Just tune the number to ensure that the second belt only activates when prioritized belt is fully occupied (whatever "fully occupied" means in your specific situation).
Essentially, for as long as the feeding belt is only partially filled, low-priority belt will always stay frozen and entire traffic will move into the prioritized belt. When feeding belt is compressed, low-priority belt will periodically unfreeze, leaking some of the items through. If priority belt is backlogged, low-priority belt will work full-time.
So much for splitting a single belt. When doing a priority split from a multi-belt bus, what you do is create a sequence of priority splitters that will shift your entire bus contents to the side from which you'll be doing the split, starting from the farthest side and ending with the actual splitter that diverts resources from the bus. This creates a characteristic "staircase": splitter + 2 wired tiles on belts 1/2, same on belts 2/3, same on belts 3/4, and finally belt 4 is priority-split to the side. This is done to ensure that a resource that is scattered among 4 belts in low amounts is concentrated on the rightmost belt before being priority-split, and this is exactly the design you see in the video linked above.