I'm not up to date on how the Turkish government is handling this but I'm willing to bet it has something to do with how they're not doing a great job at proving aid to the affected region, and they're trying to contain that information to save face.
Reading a bit it seems to be potentially that the recommendations after the last big quake haven't been followed up properly. So basically stronger building requirements, with government validation was the plan; in reality not done (properly) and "checked" by paid companies.
My first thought on seeing some of the images was that there must be some criminal negligence involved. There is one area with 20 or so similar looking apartment buildings (about 6 levels each) where every second one has completely pancaked.
I was in Christchurch, New Zealand during the 2011 earthquakes. Only 2 major building completely collapsed. One had been designed by an engineer with insufficient qualifications and the other had design issues that were previously identified but never addressed.
The media is primarily reporting good news instead of actual news. Twitter has been the best source of images, videos, and descriptions of what is happening on the ground. Search #deprem (earthquake in Turkish) to see for yourself.
Erdogan allowed (encouraged?) careless construction in many areas with complete disregard of anti-seismic building procedures. I heard someone say: "as an earthquake brought him in power, another earthquake will remove him from power". The Turkish Govt spin doctors are likely struggling to counter the wave of public outcry, hence the bans.
Including how they handled the years up to the earthquake and how much corruption influenced the country's ability to improve their housing stock with respect to earthquake resilience. I think that's the main source of worry for Erdogan and his cronies. The 'building amnesty' was a clear bone thrown at business interests at the (literal) expense of the rest of the population.
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.
Typical state excuse, "false information". Also there was growing anger, they cannot control it like the mainstream media. Media will only show the actions blessed by the state.
Paranoid government? It probably fears insturgents can misuse earthquake to spread chaos and overthrow it. I don't believe people would support them (hypothetical insturgents) in such circumstances.