Same here. I get an email when someone asks to connect but nothing else, and I get that because I want it. They have pretty explicit settings. Just checked and "Introductions and InMail" are only thing enabled. You can also adjust frequency for individual types of messages so you get batch updates.

LinkedIn is generally crappy at what is supposed to be its primary purposes, and they do shady UX stuff. But they have very granular controls for email and push notifications.