How can I update to iOS 13

It is not possible to do this because the hardware in the iPhone 5s will not support higher iOS systems.

The best that the iPhone 5s can do is iOS 12.5.6. That is the last update for the iPhone 5s.

If you want to run a higher operating system, you will need a new phone.

Even though you seem to have already solved your problem, you should know that the iPhone 5S is not compatible with iOS 13 or higher. It can be updated to iOS 12.5.6.