Oh this is easy. It's because the suburbs are perceived as having less diversity due to past decades of redlining. As somebody whose parents moved to a 'nice' suburb when I was 14, it wasn't so much nicer as it was almost exclusively white.
It may be that in many parts of the country a move to less crime is a move to 'less diversity' - but I do not think that is the case for all cities / suburbs everywhere in America.
The above comment almost sounds like one is painting their parents as making racist choices, and that other parents are making racist decisions hiding under a guise of choosing to move for safety. I do not think many people in the US are making choices for racial issues.
Moves for better schools, less noise, less crime, more family activities, and other things are generally the priority from what I have seen.
Not everyone is racist like this
Two people happily chimed in to challenge my subjective statement. Statements about how bad cities are and the issues caused by systemic lack of investment in social services rarely face much criticism.
The word choice should make you suspicious. When is the last time you heard a white person called a "hood-rat"?
In the context of this thread, which is about whether people leave cities under the guise of doing so for safety to excuse the real underlying reasons of racism, this answer isn't that satisfactory.
It's the reverse of "I have a black friend". It's "I was scared of white people too". Obviously only you can know the true reasons, but please don't think that just because you made a decision for one set of circumstances that no one made the choice for other reasons.
Instead of relying on anecdotes you can look at the data which shows that white people do leave areas in cities with more black people in them. The data is not described by pure economics. The evidence is that some people are making the conscious or unconscious decision to racially segregate.