I can tell you a lot about European (specifically Italian) urbanization, but less about American urbanization.

I do have a passion for old city maps though, and in the Library of Congress there are quite a maps of American cities drawn up in the second half of the 19th century; a period when suburban rail and city sprawl was just beginning to take shape. In this map of Minneapolis, Minnesota, for example, you can see how the city gradually becomes less dense as distance increases from the very dense area between the riverfront and the railroad; the spaces between houses becomes larger and give way to gardens, and garden give way to orchards.

Now, in Europe cities and towns tended to be older and were often subjects to restraints; premodern towns not only needed natural occurring fresh water (often rivers) but also ideally need to be on high enough ground to be easily drained. High ground is also easier to defend, and on that note, city walls were also useful in marking a city's territory. However, the area encompassed by city walls need not be completely dense. Even when expanding walls it was perfectly normal to allow for wide open spaces within the city, both to accomodate for growth and allow for small-scale agriculture to take place. For example, take a look at the expanded walls of Florence during the Renaissance, with the older medieval walls marked in red. Oftentimes, the area between the limit of urbanization and the city wall was so large that it would take the introduction of public transport to fully fill it out; look at Milan in 1860, three hundred years after the last ring of walls was constructed.

This isn't a uniquely Italian feature either. In this model of London's fortifications during the English Civil War, you can clearly see large areas of land that isn't built-up, especially in Southwark.