The clock automatically stops when the move is played. However, if you have a lot of lag on your connection, your moves can actually take longer than they appear to take on your client. You could also be suffering from some minor disconnects.
https://support.chess.com/customer/en/portal/articles/1444918-my-internet-connection-is-fine---why-am-i-getting-disconnects-
Another thing to look into is the live connection settings if you are on the website:
https://www.chess.com/blog/News/how-to-adjust-your-live-chess-connection
I'm new to this, but two of my three games I have played have been lost as a result of me running out of time. This is despite the fact that I was clearly taking less time over moves than my opponent. Should I be stopping a clock or something - like in the traditional game? But I couldn't see a button anywhere.
Thanks