I don't use iTunes, so I can't speak from experience; but I believe iTunes acts like a browser and as such mat be intercepted by the web shield streaming filter (proxy).
Depending on the errors you are getting (as essexboy asked), that may well help to determine where any conflict might lie.
There should be no need to disable all shields (as that leaves you completely unprotected), you could first try disabling the web shield and see if that is where the problem lies ?
If that resolves then it is possible to exclude iTunes from being intercepted by the web shield - avastUI > Settings > Active Protection > Web Shield - Sprocket icon (settings) > now check the 'Scan traffic from well-known browser processes only' option.