Can Verizon trace my internet browsing history?

I just got an iPhone and and If I don't use the default browser (safari) and I use google chrome, can Verizon trace my history, even if I use in private or incognito or whatever

No but the FED GOV can

Yes, most ISPs do, most people don't realize nor care though.

Most of the time they just track the actual domain name (IE. Google.com) and not the individual pages.