I am not so sure that lack of space is always the problem. It lies rather with the misguided, pro-motoring policies of governments, Department for Transport, highway authorities, police, etc etc.
In Edinburgh for example we have miles of off-road routes (tarmacadamed disused railways, the Union Canal towpath, Granton promenade, and for the past 8 years, thanks to the access provisions of the Land Reform (Scotland) Act 2003, all footpaths in and around the city, including those in parks or over privately owned land. These, however, lack controlled crossings where they meet a main road, or lowered kerbs, or quite simply short links between them, the absence of which forces cyclists to divert into fast moving traffic to attain the next section of off-road route. Our understaffed and under-resourced cycling development officer is gradually attending to the missing bits but at the present rate of progress it will take years for a truly complete off-road system to be developed.
On-road, certainly, some of the streets may be too narrow, but most of the time they are artificially narrowed by their use as linear "residential" car parks. Also some have been narrowed by the highway authority. For example Niddrie Mains Road years ago was a wide road with two lanes in each direction through a fairly poor housing area. Very properly, the highway authority sought to reduce speeding (ie reckless driving) by altering it to one lane in each direction. Unfortunately this was done by widening the pavements to about three times their normal width, rather than using the disused road space for wide, and preferably kerb-segregated cycle lanes in each direction. Now motorists are held up by 15 mph cyclists (the route is flat) and the latter are endangered by impatient and too close overtaking. I think it is just a convenient excuse for politicians and officials to claim that thanks to our towns and cities not having been flattened in the Second World War the streets are too narrow for proper cycling facilities.
That is not an attitude which we should meekly accept.