The iPhone could not be updated. An unknown error occurred (4013) - Is there still a way to solve? Today my iPhone X got stuck at a logo-flash-loop. After searching for some information with Google I tried the "voldown+volup" restart way.

The infinite logo loop did stop but I still can't access my iPhone. I saw a screen like this:



I searched again and found that there is a way to update my iPhone software without erasing my data. It says that I just need to use iTunes on Mac to update the iOS software.

Which I think is very good. So I opened my iTunes on my Mac and it really tells me that there's an update (like 13.3.1).

After clicking some confirm buttons, I ran into this screen below.


After downloading the iOS package. And everything starts to move on. I saw an error like this:



What can I do now? What is trouble for me is not just data, pictures, but I have 2 of my bank apps on the iPhone which is super inconvenient to recover if I can't access them.
it looks like you are using iTunes which is not available in the latest macOS Catalina. 
Please try the same with apple configurator 2. It might help. 

This error might have occurred because your macOS is not up to date.