Is the web application using the SMTP service on the web server itself(localhost) to send out the emails to the Internet or is it relaying the emails to your exchange server which in turn would forward the message to the Internet? In other words, what is the path you expect the emails to travel to get to an external email server?
The web app should use the SMTP service on localhost. That SMTP service should then be forwarding the message to external email servers on the Internet. If a message is bound to someone in your organization, only then, will it be sent to your internal exchange server.
If that is how you expect the emails to travel then you can perform a number of task to check why the emails are not leaving your network:
1) check the SMTP service logs and the event logs on the web server.
2)Check the relay settings for the SMTP service on the web server. It should allow relaying for emails originating from localhost.
2)Install Wireshark on your email server to examine the packets. You can save the Wireshark to a file and post it here and we can help analyze it.
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