When using Safari Web Inspector to view the Network Log, the log is reset upon navigation, i.e. navigating to a different page clears the log.
In the Chrome Develope
There is a Preserve Log option in Network.
Preserve Log
Network
The problem is that if your window is too narrow, it disappears and Safari gives you no indication that it's there. Resize your Web Inspector window to be wider.
(That was a fun one to figure out!)