when I switch to MF and slowly turn the focusing ring, nothing happens at all. The ring turns and turns but nothing else happens
It can be difficult to notice very small focus adjustments, particularly when they happen slowly. That's especially true when you've focused on a distant object with the lens zoomed out to the wide end (24mm) of the lens's range.
Try setting the focal length to 105mm and point the camera at something just a few feet away from you. Adjust the focus so that the subject is in focus, and then turn the focus ring slowly in either direction: I bet you'll see the subject go out of focus very quickly, even when turning the ring only a tiny bit.
Also, know that the EF 24-105mm f/4L has full-time manual focus — you don't have to set it to the MF setting to get the focus ring to do its thing. This is especially handy if you've set up your camera for back button focus or if you use the "one shot" AF mode, since you can have the AF system focus on the subject, but still easily make any small focus adjustments you might want.
This is my first L lens as well, so I can't test it with any others
The "L" designation really doesn't make a difference here. "L" indicates a higher build quality (weather sealing, more durable materials, etc) and often better image quality.
If you have either the 24-105 or any other comparable lens, do you also experience that behavior and do you think it's normal?
It's hard to know exactly what you're seeing, but I can certainly create a situation where it's difficult to see a change in focus. Again, it's most pronounced at short focal lengths and when focusing on distant objects. You'll still see the change eventually, but because you're turning the focus ring slowly, you need to turn it for a longer time period just to get to the point that you reach quickly when you turn the ring faster. As well, it's easier to see the change if it happens quickly.
Update: From the video you posted, it's clear that your lens isn't functioning properly. Turning the focus ring counterclockwise (viewed from the back of the camera) with the lens focused at the left end of the macro region should cause the focus indicator to change immediately, no matter how slowly you turn the ring. I'd send that lens in to Canon for adjustment.