Your cycling fallacy is…
“Our roads are too narrow for cycling infrastructure”
In practical terms, there isn't any road – narrow or wide – that can't be made safe and attractive for cycling. The solutions aren't always politically simple, but physical space is rarely an insurmountable problem.
It is true that some roads may be too narrow to accommodate cycling-specific infrastructure alongside other uses such as parking and multiple lanes for motor traffic. However, it may well be the case that cycling infrastructure is a more beneficial and productive use of road space.
To create space for cycling infrastructure, a road might be made one-way for motor traffic, or parking could be reduced. Alternatively, the amount of motor traffic can be reduced to a very low level, making the road itself a safe and attractive place to cycle.
Source: As Easy as Riding a Bike (Copyright, used with permission)