I had a similar problem with a WS5100 Ethernet shield for Arduino. It works fine plugged into the router or a Netgear 5/8 port switch, but if I connect it to a D-link 24 port switch it doesn’t work.
If you google the problem you’ll see that lots of people have reported similar issues with different makes of switches. It seems that many flavours of Arduino Ethernet are very fussy in this way.
You should probably look at an ESP8266 or NodeMCU with built-in Wi-Fi and ditch the Arduino altogether.
Pete.