> If everyone cycles you need a cycle lane at least the size of a car lane
No you don't - bikes and scooters take up far less room than the same person being sat alone in their car. The increased throughput of people would drastically improve the efficiency of the road. Also, if the people choose active travel instead of driving, then you also remove a large number of cars from the road and also reduce congestion for the people who have to drive.
New housing estates will need joining up to the existing road network, but they also need to have good public transport links to avoid the roads being just filled with people stuck in cars in traffic. Also, they should have good walking/cycling options too which is often completely neglected by developers. Obviously new developments will need some road building, but it's a mistake to think that adding roads will reduce congestion.
Ultimately, personal cars don't scale beyond a certain population density. If you aim to add more roads, the shops and facilities end up becoming more separated which then leads to more people choosing to drive to get there. The extra people driving then leads to more congestion which leads to more roads which leads to facilities becoming spaced further apart and requiring even more space for parking facilities. It's an exercise in futility.