Existing pedestrian facilities are simply unsuitable for cyclists.
My regular 3km ride to the shops takes me about five minutes by bicycle. If I had to ride in the footway with pedestrians, I would be faced with two choices - either to recklessly continue at each junction and cross the road at speed, or to stop and start like a pedestrian. Doing the former would quickly prove fatal, the latter would increase my journey time fourfold.
Most footways are far too narrow, too tightly radiused and contain substantial amounts of furniture like lighting and mailboxes, which pose no danger to pedestrians but are a potentially fatal hazard to cyclists.
It also bears pointing out that European roads were built for cyclists, not motorcars. The invention of the safety bicycle prompted the widespread development of smoothly surfaced roads. Motorcar traffic represented a minority of journeys in most of Europe until long after the Second World War. Bicycles are perfectly well suited to the narrow streets of old towns, it is the motorcar that is out of place; Many enlightened local authorities are making the sensible decision to prohibit non-essential motor vehicles from town centres, improving conditions for both cyclists and pedestrians.
There is no infrastructure problem. There are many cities with substantial amounts of cycling infrastructure but low rates of cycling and poor safety; There are many cities with practically no infrastructure but huge numbers of cyclists and very good safety.