how to see emails sent from Gmail website in Mail?

Your system is working correctly. You need to understand that the emails you send using the Gmail App in your Internet browser have NO access to the Apple "Mail" app while operating over the Internet through your internet browser (Safari, Firefox, or whatever you use.) The Sending AND STORAGE of those sent emails are being solely handled completely on Google's Servers. However, when you use the Apple "Mail" app to send your emails, then obviously that App has access to act on them and store them. That is how those two separate email systems function for your Sent emails.

Well, just so we are understanding each other correctly, if you have properly set up a Gmail email account in Apple's "Mail", then the various Gmail "Mail Folders" SHOULD indeed be showing in the left hand bar in "Mail'. To be specific, the word "Gmail" should appear in the left hand column, where mail folders reside, and it will have a small "triangle" next to it. Click on that triangle to open the drop down Gmail folders. Are you saying this is not the case?