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It may seem counter-intuitive, but in the long run it's much better to have low-capacity historic links like Hammersmith Bridge take the political pressure off road-building. The problem is that when motor traffic capacity on the old, existing network is reduced significantly, there will inevitably be projects of increasing it in other ways. Look at the cities planning to 'remove' motor traffic from their surface area and to replace it with motorways in tunnels, e.g. Hamburg and, indeed, London--see the Roads Task Force Report on pp. 168-75 here:
https://tfl.gov.uk/cdn/static/cms/documents/rtf-report-chapter-3.pdf
(There are other sections where this is discussed.)
In case you think this is pie-in-the-sky and won't be realised, quite a lot of it has already happened all over Europe. The general effect is a massive increase in the need to travel and ever-increasing economic over-centralisation. It may all look pretty, being embedded in such reports in a lot of bullshit about sustainable modes and better urban quality of living, etc., but what really drives it is *still* motor traffic increase.
I personally wouldn't mind if Hammersmith Bridge became buses, Blue Badge driving, cycling, and walking only, but it would definitely, as wildwest says, lead to extra capacity elsewhere. I doubt it would be a parallel bridge here, but someone would think of something. I know it sounds strange, but with the disaster of the Silvertown Tunnel being built, we can see that such strategic defeats are not far away at all.
Oh well. At the moment, of course, it looks as if nothing's going to happen for a while yet ... utterly ridiculous.
This would be a great outcome.