DuraikannuJeyamani-6375, To better assist you on this, what specific site extension did you install?
Kindly see if scaling up/out helps. (if you haven't done already)
As I understand from your issue description, the issue started after installing the site extension - As a quick step/to isolate the issue, you may uninstall/delete the site extension installed (from Azure Portal > Extensions blade – since you’re able to access the WebApp just fine), and then see if it helps. [screenshot below]
Just to confirm, is the WebApp on ASE (App Service Environment?) ILB ASE or Public ASE?
Capture a network trace/F12 or Fiddler trace to see if it provides any pointers.
If your app integrates with your VNet, it uses the same DNS server that your VNet is configured with. By default, your app won't work with Azure DNS private zones.
https://github.com/MicrosoftDocs/azure-docs/blob/main/includes/app-service-web-vnet-regional.md#azure-dns-private-zones
To work with Azure DNS private zones, you need to add the following app settings:
WEBSITE_DNS_SERVER
with value 168.63.129.16
WEBSITE_VNET_ROUTE_ALL
with value 1
If feasible, kindly try capturing the git http verbose output, as outlined in this doc.
Kindly let me know how it goes, I'll follow-up with you further.