The service manual will have the "float height", and tell you exactly how to adjust. The normal deal is to remove the carb, remove the float bowl. Turn carb upside down, and bend until you meet certain measurements that are in the manual (such as set float height = "bottom of carb body to top of float = 0.74 inch + or - 0.04" this is an example, I don't have the 2000 R6 manual).
When doing a hack job on a MX bike at the track (simple old MX bikes is what my experience is mainly limited to) you would:
- unscrew the cover (or bowl), if turning the gas on causes fuel to flow from the overflow tube (with bowl in place), then the float retainer needs to be bent to press on the needle valve (fuel shut off valve) without needing so much "float". If you can't get it to stop with reasonable bending, then something is wrong with the needle valve, or the float don't float

(actually another thing to check, they do crack sometimes).
- Once it's not dripping out the overflow, then you can turn the gas on while holding the bowl in place, and see how much you need to lower the bowl to get gas flowing again. If you have to lower it much, then the needle value is too "tight", and you will not get enough fuel height in the bowl. So, loosen (or bend the other way, so the float has to raise higher to cut off fuel) the needle valve retainer.
Also, you have 4 carbs on your R6, right? Is fuel coming out all 4 overflows? If so I'd have to wonder if the fuel pressure switch is not working. Sorry, I have FI R6, so I'm really not much help here

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