@Satyam Chauhan Thanks for reaching here!
The error message "net::ERR_CONNECTION_TIMED_OUT" indicates that the connection to the search service is timing out. This could be due to a firewall blocking the connection or a misconfiguration of the private endpoint. see- Create a Private Endpoint for a secure connection to Azure Cognitive Search
You can check if the private endpoint is configured correctly by following the steps in the documentation. You can also check if the firewall rules are configured correctly to allow traffic from the app gateway public IP to the private endpoint.
If the issue persists, you can try creating a shared private link to connect to the search service through a virtual network IP address instead of a port that's open to the internet. You can also check if the resource owner has approved the connection request you created.
see- Make outbound connections through a shared private link
Let us know.