Turkish families should try to flee with their children away from violent prone areas. As more protests break out and violence spread, safety of the children should be first in the parents minds. Children who are exposed to violence tend to be more violent as well as being an easy target for those who wish to cause harm.
Yes, families with children should try to leave areas that are prone to violence, because the safety of the children should be the most important thing. These children can look for safe harbor or even refugee status in the United States or somewhere else. Staying where you can get hurt is not a good idea.
As a parent I would desperately try to remove myself from any location that was prone to violence. I think countries that surround these areas should help and provide assistance to families so they can leave these areas. If violence is proven to depopulate a country, it very well may limit the amount of violence that is used in the future.
Families in the protest areas should seek out safety if there is a chance that protests will turn violent. Parents have a responsibility to look after their children and guard their safety. In order to keep children safe, whether the parents support the protest ideals or not, they should look out for their own first and consider seeking shelter elsewhere.