If you mean a ribbon strip along the bottom of the screen, then no this isn't normal as this option isn't enabled by default. See image1 example, whilst this is an alert screen not just notifying of files scanned, they look very similar.
Open the on-access scanner screen - Standard Shield - Left click on the avast 'a' icon, click the Details... >> button if shown, select the Standard Shield provider > Customize > Advanced - 'Show detailed info on performed actions' this option is off by default, so it would appear you have been tweaking avast without knowing the effect it will have.
This function is also on the Web Shield and Internet Mail providers so you should also check those.
Edit: changed image1 to show correct list.