I have win 10 on laptop. I have only avast and windows defender but win defender is off. I don't have internet connection issues.
By internet connection issues, that could mean problems connecting with the avast installation servers, not necessarily on your system or ISP.
Still when I click shut down. It says avast antivirus installer is on. Don't shut down your computer. But what happens if I click nevertheless turn off?
I wouldn't rush to shutdown if the avast installer/installation is really still running. Potentially it could corrupt your installation.
What avast processes you you see of avast running in the Task Manager ?
If you check the Windows Security Center what AV does it say is running ?
If you hover your mouse over the Avast Tray icon, what is displayed ?