The history of the suburbs in the United States has been tied very strongly to the history of transportation. The more that transportation has become available, the more that suburbs have arisen. This process of transportation advances leading to suburbanization took off in a really big way in the 1950s.
Before the 1950s, suburbs had existed on a relatively small scale. They started with the rise of street cars and local rail systems. These transportation technologies made it more possible for people to live some way from the city center while still being able to get to work on time.
Suburbs really took off after World War II. They might have been able to boom earlier, but cars only became really widespread in the 1920s and that decade ended with the start of the Great Depression. With the Depression and then the war, there was not enough spare wealth to allow for much suburbanization. With the end of WWII, that changed. The government engaged in road building and it subsidized housing loans. The new-found wealth and the ease of transportation allowed people to move to suburbs. This was the time when suburbanization truly began in the US. Since then, the trend towards suburbanization has continued and most Americans today are classified as living in suburbs.