In the case of protests etc I'd understand the goal behind it from certain actors involved, but why in this case?
Despotic regimes hate to be criticized. Or anything that might make them look bad, including how they handle an earthquake.
Thousands of buildings collapsed killing lots of people inside them. Once the dead have been buried there will be calls for why nothing was done to improve the housing stock and in a de-facto dictatorship the buck stops with the dictator. Erdogan isn't going to be able to weasel his way out of this one without claiming another fake coup.
And that's before we get into the fact that for obvious reasons Turkey isn't able to handle the aftermath of this alone, and that makes Erdogan look weak. The fact that they are happy to switch off Twitter even though it is saving lives by allowing people to be found under the rubble is telling: even now the lives of the Turks matter less to the regime than their own image.