You don't need to update your Flash player but disable it for the site (only videos on v2 use Flash).
I'm not going to say Flash is your problem but I have seen instances where it can cause lag. Also, look through the following; it is about disconnects but some of the suggestions can help with lag too.
https://support.chess.com/customer/en/portal/articles/1444918-my-internet-connection-is-fine---why-am-i-getting-disconnects-
I have been facing quite huge lag issue even the net speed is 4mbps. is it not adequate??? all free members got this issue? lost many games by lag. i updated flash player too. let me know if got any solution guys plzz...