Time Machine is backing up itself? I have a MacBook Pro 13 inch 2017, 2 USB-C ports, MacOS 10.13.6, with about 90 GB content, 120 GB total HD. It gets its first backup onto a pristine, just reformatted, 500 GB external drive, via USB. The drive is connected via a USB to USB-C connector (in case this could be an issue).

The backup starts fine, then it goes past 90 GB and it keeps ballooning until it "runs out of space". I made a small clip of the window  that displays this process which I can't show here but here is the display on the first frame: "Backing up 325.82 GB of 358.40 GB" and on the last: "Backing up 326.00 GB of 358.60 GB". I can see, looking at the screen, as both numbers just keep climbing up and up. 

There is nothing else connected to the computer. 

What's going on? It seems the backup goes into an infinite loop.
It would be very strange to have TM back itself up... That said, we can check! Go to System Preferences > Time Machine > [Options...]. The drop down pane that appears has "Exclude these items from backups:" If your external drive is not listed there, it might actually be backing itself up. Click the plus under the list box and try to select your external drive. If it now appears in the list box, click [Save] and try to backup.