Just had the same problem with the update progress bar on the phone stuck at the very beginning of the update process. Note I am updating an iPhone 13 pro from iTunes version 12.13.4.4 on Windows 10.
First of all, to get the iPhone to respond, I used the sequence to put the phone in recovery mode. That is press and release volume up, quickly followed by press and release volume down, then press and hold the right hand button until the screen unfreezes and the phone goes into software recovery mode.
I then opened iTunes on the pc, and connected a USB cable to the phone and PC. iTunes detected the phone and offered me two options. Either to attempt to recover the iPhone or to reset the phone. I wanted to keep all the settings and documents on the phone, so I chose to recover the phone. iTunes then restarted the update process and this time completed correctly. All of the settings, apps, documents, photos etc had been completely preserved.
Not much more I can add, well done to Apple for making a recovery option that worked for me!