If you download and run the free program TCPView from Microsoft/SysInternals it will display the processes making network connections. Then you should be able to see the process that is connecting (via avast) to the network.
I thought about that before replying and my thought was …
If the Webshield is stopped it will cause the connection to simply terminate and may well leave no download going. The poster would probably need to restart the system with the Webshield still terminated to see the unwanted process restart its download.
The browser is unlikely to be the issue here … the user suspects that some (as yet unidentified) process is causing the download to occur. If this is the case it is most likely being triggered at start up - not by the user initiating the browser.