Private Browsing mode in Firefox won't change how you are connecting to websites and it doesn't stop your ISP from seeing what you are doing on the internet. Essentially, Firefox Private Browsing mode just makes sure you clear your history and other browsing information when you close it. It's primary use would be so that people who share your computer can't see what you were doing.

It wouldn't have the effect that you are describing.

The only thing I can think of is that maybe Private Browsing mode is using a stricter setting for the Enhanced Tracking Protection feature, which may be blocking some scripts from loading in that mode.

It's also possible the problem is caused by an add-on that is not active in Private Browsing.