It's difficult to give advice on the performance issues you are facing without having further details on how your Elasticsearch infrastructure is architected and built. Could you provide some more info on what setup you are using on both AWS & when building on-premise?
Also, it would be useful to see your elasticsearch & kibana configuration files (please don't forget to remove any secrets though, if you have any!).
Also, the request timeout can be increased in kibana.conf file. Just added elasticsearch.requestTimeout: 90000 to /etc/kibana/kibana.yml and check if the error goes away.
Thanks
Rashmi