Check the %usage of the output units - if all are maxed out at 100% then its a good bet that something is not getting to where it needs to get to. For fast moving goods its often useful to put in 3 inputs and 3 outputs, even as many as 4 inputs can be practical because if I'm not mistaken the outputs are more capable than the inputs as they train up more quickly. Basically the 4 corners would be inputs, the horizontal 3 are outputs and the two left over either side are storage, all linked accordingly. I think 4 outputs on 3 inputs will suck the inventory dry too quickly if I remember correctly and I believe 2 outputs on 3 inputs works quite well for wool/cashmere/seasonal agri products as it allows you more space for storage to build up inventory for when the wool/product is out of season.
Its been a while since I checked this though, so other layouts may be better, but that's the basic theory at least. I also like to run my warehouses for a month to give them a head start on building inventory before I allow clients (either my own or AI) to get access.
Pro tip: It's also very easy to suck animal product farms dry with warehouses and thus build up their expertise more quickly (the units are constantly working and learning), which can help with quality for production (or even just sales to AI). Product quality blending is also easily done using warehouses. The whole warehouse thing is a little bit left pocket/right pocket tbh, but it works quite well overall for managing your empire.

As far as I know, when there´s a strong pull from the warehouse the demand bar grows over the supply bar, but it tends to last just a few days. You´ll probably see a more balanced ratio after a while.
If you see that persist and the warehouse storage boxes are still full, open a second sales dept. box.

I have a question regarding the supply bar on an individual warehouse. Does that indicate the actual fulfillment rate or does that just indicate the supply replenishment available vs the current demand?

Here is a break down of what I'm trying to figure out: Say I have a warehouse with full inventory of a product. Then say I set multiple retail locations to pull from the warehouse and the supply bar sinks below the demand bar... does that mean that those orders are not being fulfilled even though there is inventory, or is that just showing that the demand exceeds the supply coming in (so if I let the inventory run out it will bottleneck).

Thanks to anyone who can answer this. I can clarify more if needed.