That was already happening. It was called red lining, and it meant as a bank you'd only approve loans for black families if they lived within the red line drawn by the bank on the map denoting the designated black neighborhood. Banks would also do something called blockbusting. They'd send panicked pamphlets to white people in urban neighborhoods, alerting them of black people moving into their neighborhoods and the damage that this could cause on their home value (the only asset many people had at this time).
Now suburbs were being built because in 1946, suddenly there were millions of young men with a huge government subsidy to buy a house. Banks would not approve mortgages for black people in these areas, so they became predominantly white. And as a white person you are happy to be in the suburbs where you are free of all the perceived fears from decades of racially charged propaganda, and surrounded by people who also have a white boy and a white girl, a new ford, a dog and a cat, and who go to the same protestant church. It was sold as a utopia, and in the case of Detroit, white people bought that narrative so hard the city lost 1.5m from its tax base and collapsed in fewer decades than it took to grow as a metropolis in the first place. To date, Detroit has lost over 60% of is population from its all time high in 1950.