This is precisely why cycle paths and lanes are less safe than riding on the road: a left turner on the right and a vehicle on the left going straight ahead, both of whom are adamant they have the right of way. Most accidents occur at junctions, not between junctions, so cycle paths reduce the minor problem, and increase the major one.
This is part of why we don't get nice things. They may both be "adamant they have the right of way" but the turning vehicle operator is wrong and we should tell them they are wrong, rather than pretend they are correct! It is very disappointing to read someone on a cycling forum trotting out these old motorist myths.
We should, of course, lay junctions out so that the cyclist has more warning to spot and avoid an incompetent motorist: either by bending the carriageway right, away from the cycleway, so that the paths cross at right angles; or where there are traffic lights, by holding their left turn while the cycleway has a green.
If motor lanes were laid out the same way cycle lanes, nobody would have any difficulty whatsoever in seeing the crass stupidity of it:
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Except that exact layout happens at various places! Even better, there are some level junctions on fast/busy roads where the lanes are: left turn, right turn, ahead, ahead. I believe that's because it's considered undesirable to get turning lorries into lane 3/4 and wait for a gap in oncoming traffic. The conflict is prevented by the junction traffic lights not showing green to both right and ahead at the same time. We could do this — and much more — for cycleways, if highways departments were willing to install better signal controllers instead of the simple stupid ones widespread in the UK.
(There are, of course, many junctions where the right turn lane is on the left and the roads cross using a bridge to keep them apart.)
[...] If there's any room to argue about who has the right of way, the road design is plainly at fault.
So every single unmarked junction and all those modern shared spaces are all "plainly at fault"? What about crossroads where right-turning vehicles arrive at opposite minor arms simultaneously? Plenty of room to argue who goes first there.