Not at all.
What’s the sense of “in private browing” then?
It’s a sham?
Private mode isn’t a sham, but it is limited in the privacy it offers you (not much). It just makes it harder, but not impossible, for a web site to keep track of what you’re up to.
Running some sort of encrypted session to a collection of uncrackable proxy servers I think may be the closest way to obtain some sort of true privacy for your web traffic.