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no one does it but TFL Cycling
Not quite; two-stage right turns like this are modelled on the example of Denmark. They are very cyclist-unfriendly and designed to increase motor traffic capacity (rather than providing a proper arrangement for making turns). I missed that that was there (probably because for heritage reasons, e.g. not being able to create markings on the footway, they couldn't make it very obvious). It's yet another regrettable feature of this appalling scheme.
As SwissChap says, the arrow is pretty unclear. Also, you have a cycle track opposite, in Bridge Street, with a prohibition to turn right. There is no indication there at all that in order to perform a right turn, you're meant to mount the footway and wait at the stop line painted there. As to make this possible this section of footway has to be shared-use, the area of shared use should be demarcated by appropriate paving so partially-sighted people can detect it, but I imagine they couldn't do that there for heritage reasons, either (it does look rather ugly).
Another thing that's always annoyed me here, unrelated to the scheme, is that there's no pedestrian crossing from the west side of Parliament Street to the island in Parliament Square. That's been the case for as long as I can remember.
‘Legally’ as a cyclist coming off Westminster bridge on the cycle lane turning right onto Whitehall you have to make a two stage right turn by waiting on the pavement on Parliament Square corner. It’s nuts and no one does it but TFL Cycling 🤷🏼♂️
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