Unfortunately avast has no control over your connection speed, I'm also on dial-up and I see no difference at all to what my connection speed is. Some might see a difference in their download speed (high speed broadband perhaps) but that has nothing to do with connection speed.
So I would say there may be some sort of conflict but what that might be that would effect connection speed is beyond me.
The avast 'a' icon (I don't know if that is what you mean) only rotates when it is scanning something but there shouldn't be that much to be scanned in the page that you are connecting and that again shouldn't have any impact on the connection speed. So I don't know how the coincidence is there when you stop avast it connects at a higher speed. You shouldn't have to stop avast completely try terminating the web shield provider and see how that works.
The web shield provider is what scans web pages and that usually has little impact on download (browsing) speed and less on dial-up as it has so much more time to scan content as the download speed is relatively slow.
What is your firewall ?
What Operating System are you using ?
Have (or did) you another AV installed in this system, if so what was it and how did you get rid of it ?
What other security based software do you have that might have an impact ?
All of the above however, really shouldn't have any effect on how your modem establishes a connection nor the speed it does it. In the three and a half years I have been on the forums this is the first time I have seen anything like this.
