You would think that you should be able to trust an anti-virus provider that has been around for a long time and that the provider should be aware of safe practises and not expose you to the dangers of the "Wild Wide Web"
However, an investigation by PC Magazine and Motherboard found that Avast may actually be tracking, recording and selling the Internet browsing histories of their users! Although the data is purportedly anonymised, the investigations found that this may not be the case as some of the data recorded may actually allow for users to be "de-anonymised". While the article does not state exactly how this can be done, I suppose it may be through the embedding of user and session IDs in the URLs.
If you are using Avast antivirus and you wish to stick to it, perhaps you should do an uninstall and reinstall it to state specifically that you do not wish to share any data. Alternatively, you may choose to use another free antivirus product. Personally, I use Comodo Internet Security as their license is open enough to be used both for personal and commercial use.