Greetings Tom Willis,
Welcome to the Apple Support Communities!
From your question, it sounds like syncing completes without syncing any apps from your Mac’s iTunes library to your iPhone 5s. I know that syncing issues can be frustrating, especially if you have already looked for information in Apple’s resources and have not found a resolution, but I do have some troubleshooting steps that I would like you to work through.
1. Connect your iPhone to iTunes and select it.
2. Navigate to the Apps pane by selecting Apps from the Settings section of the sidebar.
3. Make one change to the Apps column. Add an app to sync to your iPhone by selecting Install or remove an app from your iPhone by selecting remove.
4. Select Apply in the lower right corner of iTunes.
This should start a sync, including applications. If your apps sync as expected, you can return to Step 3 and add/remove the change you made and then select Apply again to give you your correct apps. If the apps still do not sync, and the change you made was not reflected on your iPhone, please let us know.
Cheers