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
This was just added to WebKit last week: https://trac.webkit.org/changeset/211451
You can expect it to make it into a Safari Technology Preview build soon. Once it's in a build, you can toggle preserving logs on navigation in the Settings Tab.
There is a Preserve Log
option in 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!)
In Safari 11.1.2 this is a setting on each tab under the checkbox "Preserve Log"
Note that if your Web Inspector window is too narrow to show all options, you'll have to expand it until it is wide enough to display the Preserve Log option, as mentioned in an answer by ShortFuse.
In Safari 11 this was in the Settings panel under "Network: Clear when page navigates", and it's sister setting "Console: Clear when page navigates".