In other countries the population will unite after natural disasters, and leadership is usually given credit for the response as long as they continue support for local response efforts set up in the immediate aftermath. Why would popularity of the leader drop so quickly due to an earthquake? It's been what 36 hours. At this point almost all committed resources will still be highly local...
Potentially because they cut off social media essential to life-saving efforts, and focus on managing optics rather than providing concrete help? Just a stab in the dark, but who knows.
Well, either Erdogan knows he's deeply unpopular, so he/his lackeys has to censor all the bad news and criticism so he/they can look a bit better, or he's afraid he's deeply unpopular, which means a bit of self-awareness of what a raging asshole dictator but useless leader he is... Fun times!
A leader that actually cares but is overwhelmed would probably not be afraid of admitting that, s/he could rally the population together to overcome the tragedy despite their lack of resources. I think Dubya managed to convey this with 9/11, no one really knew how bad decisions by his administration lead to the threat being unnoticed. Meanwhile with Erdogan, he's been the boss for many years, the public probably thinks: The government's not able to rescue people effectively? Well, that's because it's been a kleptocracy for a long time.
Elections are meaningless if you can print millions of untraceable mail-in ballots and do not enforce rigorous ID requirements. Furthermore, elections are even less meaningless if you can imprison the opposition or destroy them in media under false and contrived pretense.