Security yes, privacy not especially.

PGP lets you encrypt the messages and sign them to digitally prove you sent them.

It doesn't help with the problem here which is that the metadata of who you are (the IP used to log into the webmail and the email address of the sender) and who you're talking to (the email of the recipient) and when (timestamps etc.) were able to be leaked.

In fact, depending on the implementation, PGP could be considered slightly worse for privacy because you'd have the added identity proof of the message having a signature that only you could create with your private key (although that's encrypted, it's a stronger identity proof than the sender email address). It also generally leaks the recipients' key IDs too (although that's configurable) PGP is great for accountability, message confidentiality and non-repudiation. Not so much for privacy. For that you'd need other systems.