If they work, that is. Same goes for any policy, investment, or change, really. Whether they work is in turn somewhat dependent on whether they are popular, of course, but what determines whether they are popular is different before and after implementation. Before implementation, various campaigners, interested parties, and media sources will be fighting to persuade us what a great / terrible thing the intervention will be, and before implementation an intervention might be unpopular because people have been given a false idea of how bad the problem is, what exactly the proposed solution is, and how likely it is to work. Once they see the intervention working and how much better the world is when that problem has been solved, the opposition fades.
The HoL Sci & Tech committee’s report on evidence-based behavioural change cites the example of the ban on smoking in public places, but a more relevant example might be the London Congestion Charge. Various powerful interest groups raised opposition to the charge in our Motorism-infected media, spreading scare stories of chaos and failure while playing down the problems that the charge was intended to solve. It was unpopular because people were told that it would make getting around more difficult. The reality was that the charge paid for or otherwise enabled a massive improvement in bus speed, frequency and reliability; Oyster; and better streets. For most people, getting around got easier and the streets got nicer. The Congestion Charge was unpopular when people trying to imagine it, based on the misinformation they had been given, but became popular when they could see the reality.
(Sadly this lesson has been missed by Edinburgh, Manchester, and New York, whose politicians rejected road charging because polls showed it to be unpopular. It’s alright for that London, say the newspapers of Edinburgh and Manchester, but they’ve got better alternatives to the car than us — failing to acknowledge the role of the Congestion Charge in providing those alternatives.)
The same applies to bicycle infrastructure. Councillors and politicians thinking about investing in bicycle infrastructure, perhaps even at the expense of a few parking places or a taxi rat-run, shouldn’t spend so much time worrying about per-implementation polls or how large a segment of the population are existing bicycle users. Opposition goes away once people see the reality — that there is no new parking or congestion apocalypse, only a new, better, option for making journeys.