Badly. There's this thing called the "third party doctrine" which basically says the 4th amendment doesn't apply to information you give willingly to a 3rd party, so the government can just ask the 3rd party for it with no need to get a warrant first.

There's some laws from after this doctrine was established that attempt to rein this in by explicitly extending the 4th amendment, so like that's why the government needs to get a warrant for a wiretap, but without a specific law passed by congress, the third party doctrine applies.

I tried to google if there's a 3rd party doctrine for mail services, but I haven't been able to find anything yet.