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There must be some trick I'm missing. When you bleed brakes you really don't want the bars to be flopping around. How do people keep them steady? My best effort is a hooked bungee around the shifter and wherever is appropriate on my workstand but its not really that reliable. Any better hacks?
If you're that bothered then just over-tighten your headset. I've never found it to be an issue.
We bought a tool from Planet X when they were selling them off but I'm sure I could have fabricated one from a coat hanger
A bit like the link but 1/4 of the cost
https://www.amazon.co.uk/Park-Tool-HBH-2-Handlebar-Holder/dp/B000QSXPL2
strap holding the wheel to the downtube, if that can't move nor can the bars.
If you're old enough, you might have some toe straps lying around. Wheel to the down tube always worked for me.
Two rubber bands around front wheel valve stem, stretched Either side of tyre and hooked over DT bottle boss bolt.
I don’t bother my arse though.
I always take the wheel out when bleeding the front or rear brake to remove the pads and get the spacer into the caliper
If the bike is in a stand, make sure the front of the bike is lower than the rear and the bars won't flop around. The'll still be able to move, obviously, but it's a lot easier to manage.
If that's not enough, fashion a handlebar holder like this out of old coat-hangers:

https://www.chainreactioncycles.com/mobile/park-tool-handlebar-holder-hbh-2/rp-prod17706
Expensive for what it is but works well
Park do a handlebar holder you can buy or copy the idea.
I also sometimes strap MTB bars to the top tube (depending on the bike and brakes)
Old inner tubes from the ends of the bars to the back of the saddle. Keeps everything nicely tensioned.
That HBH-2 felt like one of my most frivolous purchases ever. OTOH it’s also ended up being one of my most used tools
Pointing the bike downwards makes sense, I usually have it pointing up purely so that I don't need to bend over as far...
HBH2 is simultaneously crazy money for what it us, surprisingly cheap considering it is Park Tools Blue, and also perfect to copy with a coathanger, if I can just find a wire one!
As mentioned, hang it nose down, sorted. Or just lean it against the wall with the other brake tied back