You start by fixing the problem of people sleeping on benches and in tents. Then you go to those in cars, then those crashing on a couch, then those living 8 to a house, then families with small places, and so on. What the problem is keeps expanding until the resources allocated to it are spent.
In your example, (it seem to me that) is not that the problems is expanding (getting worst) but it is reaching (helping) more people.
Often, the homeless programs provide aid to the homeless with food and clothing and health care, not necessarily to make them not homeless.
It is much harder and expensive for homeless programs to create shelters or homes. It is also difficult if not illegal to force people into housing.