When you first plug in an Xbox controller, you will see a pop-up for the game bar if you click the Xbox button. To turn off this feature so you can use the steam overlay in big picture mode do the following:
In Steam, press Win+G or Xbox button to open Game bar (choose "Yes, this is a game.")
Open the settings (cog icon) on Game bar and uncheck "Open Game bar using [Xbox logo] on a controller"
Uncheck "Remember this as a game" and Game bar should close
Now Game bar won't open when pressing the Xbox button anymore (seems to be global) and the Xbox button works as expected again in Big Picture mode. To open Game bar later, press Win+G.
Source: here
