No it doesn't send every site you visit back to avast.
It also doesn't send malicious sites back to avast (that would be massive with 400 million plus users.
In simple terms the web shield, scans the content being downloaded before it arrives on your system, if it is malicious then it would break the connection to prevent infection.
Some data on the alert may well be uploaded site location and what malware detected for updating signature detections/protection.
