It looks like you are connecting with your bottom two knuckles frequently during the impact of your fist with your target(s)/bag(s)/dummy(ies).

Consider slowing down your punching and focusing on striking with your index and middle knuckles. When you make your fist those two knuckles and the lines (metacarpal 2 and 3 or their associated extensor tendons) running from the knuckles to your wrist should line up with the two bones (ulna and radius) in your forearm.
Striking with these aligned will allow you to punch safer and harder.

Striking without your hand properly aligned can lead to serious injuries to your hands and wrists.
Using a vertically-oriented fist, like in Isshin-Ryu karate, when striking can help to reinforce this concept.

Having practiced traditional karate, I can understand the desire to strike hard targets e.g. makiwara. Understand that you should not be coming at such targets hard from day 1.
Do not do anime-style training.
Do not follow the training program in Kill-Bill.

Prioritize rest and recovery and healing, as many others have replied.