Safari is getting a download manager in iOS 13 which is due to release sometime later this fall.
iOS 13 Hands-on: Safari Finally Gets a Download Manager
New Safari Download Manager feature in iOS 13 will make you more productive
I don't know what iOS version you're using. I tested on iOS 12 and Win7 with iTunes installed. Just plug the iPhone with a normal USB type-a to lightning cable, then type the password to trust the computer on my phone.
The iPhone appears as a portable device in My Computer and a similar path Internal Storage - DCIM - 100APPLE I can find the videos. Simply ...
To find the program that is using this service, I have used the Energy tab of the Activity Monitor. Identify the highest energy-consuming app. Much simpler than spindumping as suggested in lowrin's answer
I have done this with Safari Version 12.1.1 (...