David Hembrow is an Englishman who lives in the Netherlands and writes a blog called A view from the cyclepath. He also has a You Tube channel about cycling infrastructure. He’s very good at making observations about people, transport, and infrastructure and drawing conclusions from what he sees.
This video is a very good one about cycling on pavements in Norway. It explains very well why in busy areas it’s best for pedestrians and cyclists to have their own space.
rachel.blog 
That bridge looks like one we have in Lewes – “mostly” pedestrianised area which is in the middle of a popular cafe/shop area. At busy times I end up getting off and pushing, which is frustrating and it is lucky I tend to get back home after the busy times. I could ride very slowly but it’s so possible that a pedestrian could unexpectedly swerve out, not knowing that you are there.
That sounds exactly like the problem in the video. Cyclists shouldn’t have to get off and push because, as he says, it makes cycling less efficient and less attractive then people won’t do it. If we want to increase cycling modal share as much as possible we have to make it as attractive as possible.
Yes that’s true, it’s frustrating but I put up with it in the grand scheme of things but there must be many people who are put off. I also don’t like the feeling it gives me that I am imposing as a cyclist on the pedestrians.