Trial and error and I now have Avast running with Online family and able to browse if I disable Avast Web shield?
By disabling the Avast Web Shield, you are now exposing yourself to potential malware since this is a main AV protection. You need to decide what is more important, monitoring your kid's activity with Norton or using an antivirus (Avast).
I looked into the limited info. available on the Online family and you also need to download another Norton product for it to run. I can see this potentially conflicting with Avast's web shield when you are online. You might do better having a firewall with HIPS protection where you can easily see via history log where they surfed as well as blocking web sites from being connected to your machine (Online Armor Premium for example), that does not conflict with Avast if you put both products in trusted exclusions and you will have better protection.
I realize you are concerned with where your kids are surfing, but you can educate them on responsible Internet surfing while maintaining protection for your machine as well. We can give you more tips if you like, however I realize this is your machine and your life (with your kids). Please let me know how I can assist you. Thank you.