Ethernet Shield - Multiple simultaneous connections cause TCP reset
I have received a request from the OP to move this to Network Engineering because it got no answers. I'm going to decline this for the moment because I doubt they will be familiar with the Arduino Ethernet shields.



The low-cost shields have a known limit on the number of connections they can handle simultaneously (no doubt limited by available RAM). From the page Arduino Ethernet Shield 2 we learn:


  It supports up to eight simultaneous socket connections. 


You don't seem to be getting that, which makes me wonder if it is a code issue. It might help to post your code. If you make code that "blocks" while handling the first connection you may not accept the other ones in a timely way, leading to a reset.

Please post a Minimal, Complete, and Verifiable example.

I did a post about a tiny web server for the Arduino which uses a state machine to handle incoming data without blocking.



You may also be able to help yourself by using the HTTP header that requests that the connection be kept open, eg.

Connection: keep-alive


That might stop the client opening new connections in quick succession.