The fact is, the core system files of iPhone are corrupted during the update. Restoring can't be avoided. In fact, your iPhone is asking you to restore it from iTunes and 90% of in these situations, the data is already gone. Like I said, in rare cases iTunes may be able to Restore and Update without losing the files. But if this doesn't work, then you have to restore it. It doesn't matter where you take it, even to the Apple store, they are going to restore the iPhone.
I know you are looking for a way to restore your files back, but believe me there is no way to prevent the restore. The restore is done to restore the iOS software.
First download and install latest iTunes on your computer from here : https://www.apple.com/itunes/download/
Then connect the iPhone to your computer with the USB Cable.
Open iTunes application on your computer.
It will recognize the iPhone and ask you to restore or update. So first, click Update. Then follow the onscreen instructions to do the update (update is also a restore process but this may preserve data while restoring). If the iOS is completely corrupted, then this process will fail. That means everything is already lost from the iPhone.
So disconnect it from your computer. Then reconnect it and this time click Restore. This will restore the iOS software and put the iPhone to working condition.
Once the iOS is restored, follow the onscreen instructions on the iPhone to setup it. While setting up, you will get a option to setup it as new iPhone or Restore it from iCloud backup. Click Restore from iCloud backup. If there is any backup made, then this will restore all the files from iCloud.
Remember, it may take a while to restore the iPhone (the iOS firmware has to be downloaded, you will see the Download button at the top right corner of iTunes screen. Click it to see the time remaining)
If nothing is recovered, then there is a third party application that you can try to get your data back. The name is***** for iOS. Get it from here --> https://www.wondershare.com/drfone-iphone-data-recovery.html
Install it and see if it can see your data on iPhone. Try the trial version of that application
Please let me know if that didn't work, so I will continue to help you until we fix the problem.