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
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".