While testing any site using BurpSuite, if we send any request it will use BurpSuite HTTP "Proxying through BurpSuite" but still we are not fully anonymous, it will still use our original IPv4 address to send requests. So I tried to change my IP address using VPN "Extension" on browser but unfortunately while using VPN, BurpSuite stopped it's proxy and was not able to catch/send any request.
How is it possible to change the IP address and use BurpSuite proxy as well?
What you're referring to as Proxying through BurpSuite is how Burp allows you to sniff and manipulate traffic, or perform what is called a Man-in-the-middle (MiTM) attack .
Burp Proxy lets you intercept HTTP requests and responses sent between Burp's browser and the target server. This enables you to study how the website behaves when you perform different actions.
https://portswigger.net/burp/documentation/desktop/getting-started/intercepting-http-traffic
This action isn't about hiding the identity of the request initiator from the server, but rather read or manipulate the content of the request or the response, in order to collect data or try to make one of the applications (the server or the client) do something that it isn't intended to do.
What you're looking for is setting up an Upstream proxy server.
These settings control whether Burp sends outgoing requests to an upstream proxy server, rather than sending them directly to the destination web server.
Which can be configured under the User options -> Connections -> Upstream Proxy Servers section in Burp.
