I had exactly this problem when trying to upload some videos from my Mac to my new iPhone X for the first time a few days ago! Although everything is right up-to-date on both my iMac and iPhone (macOS 10.13.3 with iTunes 12.7.3, and iOS 11.12.5) I could not find the new TV app anywhere on my phone. It just wasn´t there. But it should be a part of iOS. I finally worked out what to do:
1) On your iPhone go to the App Store and download Apple´s free TV app. Just type “TV” in the search field, and make sure you get the right one! It looks like this:
2) On your Mac open iTunes and import into iTunes any videos you want to upload to your iPhone. You´ll then find them displayed in “My Videos”.
3) Connect your iPhone to your Mac with a USB cable as usual, and with iTunes open you can now sync any video in iTunes to your iPhone, after making sure they're selected. The videos can then be found by opening the TV app on your iPhone and selecting “Library” at the bottom. They may take some time to appear, so be patient!
The trick in all this is that you have to get the videos into iTunes on your Mac first, then sync to your iPhone. Once you get your head around all this it´s easy to do and fast. Apple has just confused the issue in typical fashion by calling the new app “TV” — a better name, IMHO, would be “TV and Videos”. Hope all this is understandable and that it helps you! Cheers, Ian.