Camera flash produces unwanted artifacts on the video
This is actually a collateral effect of the concept of rolling shutter. Most video devices do not expose all the frame area at once. They expose starting from one side and (up or down) and move from there. (The example has the camera rotated 90° so the effect is on the sides of the frame).

This can be either a mechanical element that moves or rotates or an electronic system.

As the flash of a camera is a really fast burst of light, it can sometimes be captured in one zone of the frame, by the time another part of the frame is streaming the data, the flash burst is over, so this zone of the image is transmitting the non-illuminated scene.

On still frame photography, this is the reason you have a minimum sync speed.

Frankly, it is really hard to avoid this situation in an environment such as the example posted, because you can not sync your camera to that flash.

On a controlled environment, like lighting in a movie, you could add the lighting in post-production, making one full frame white, and probably some other just a bit exposed.

Another way to control it is by using a slower burst of light. Rather than using a flash that lasts about a thousandth of a second (which disappears really fast), you use some system that flashes on tenths of a second. This way the burst of light has time to be captured by the shutter of the camera.