The third item in the "What doesn't work" section of the Star Trek: The Next Generation writers/directors guide notes that Starfleet isn't the space police. They're not there to maintain and enforce a particular vision of order on the galaxy, or even on worlds affiliated to the Federation. (See also Starfleet's military status, or lack thereof.)
As detailed on Memory Alpha, some colonies in the Next Generation period seem to be pretty autonomous from the Federation, even if they were originally established as Federation colonies. Much as alien worlds who join the Federation maintain some self-determinationalien worlds who join the Federation maintain some self-determination, the objective of some colonies might be to become independent worlds, rather than being ultimately governed by the Federation.
And sometimes in space, things don't go according to plan — Starfleet isn't omnipotent, and the Federation doesn't have enough starships and diplomats to send them to every colony that has problems. Once Turkana 4 declared independence, which seems to have happened some time before society completely broke down, it ceased being a part of the Federation. Starfleet would be invading if they tried to take control of the situation, and that's just not what they do.
A Starfleet ship did attempt to re-establish contact just nine short years after the colony cut themselves off, and for all we know, Starfleet provided some degree of evacuation back when things were going badly. Some people (like Tasha Yar) still made it off the planet and into Starfleet itself; while others (like Tasha’s sister Ishara) decided to stay, as they supported one of the factions vying for control of the colony, and didn't seem to have any interest in Starfleet assuming control of their colony.