Why hundreds of years? American and Canadian cities became car-dependent sprawls in the span of a few decades. A concerted effort to redesign them would not of course have results in months, but 10-20 years are enough to completely transform them.

Also, when we are talking "redesigning", don't imagine SimCity like buldozing and rebuilding. It can start with doing away with zoning regulations mandating single family homes everywhere, doing away with strict commercial/residential zoning, doing away with parking minimums and allowing people to sell off parking lots for development. Then couple this kind of libertarian-style deregulation, with socialistic-like public investment on public transit and amenities (that should be much cheaper for denser neighbourhoods). In the US and Canada, good public transit will probably mean trams and trolleys, or (sigh) buses. Finally, establish norms that require good cycling infrastructure on any new road being built and any old road being repaved. It won't be too long for change to happen.

Finally, one more thing: E-bikes and e-cargo-bikes, along with quick infrastructure fixes (e.g., blocking off some roads or blocking off one direction in stroads with islands to make them transit and bike-only) are a much much better stop-gap solution than electric cars. The vast majority of car trips are with only a single person. Why haul a few tonnes of steel and plastic around? Instead, ebikes need much smaller batteries, and cost only a fraction of the cost. They are fast, and comfortable and can cover larger distances and you don't need to be sweaty when you get there.