Betta can breath from the surface as well as through their gills. They get air from the surface and their labyrinth organ allows them to absorb the oxygen into their bodies. Can we get a closer picture of his injury? I will need you to answer a few questions, it is very important that you do so before we can provide any immediate instructions. For right now I suggest another immediate 50% water change of conditioned water. Make sure the water has been treated with water conditioner and is as close to the tank temperature as possible. Pour the water in gently to prevent your betta any undo stress. Make sure you DO NOT USE anything with -afix (melafix, bettafix etc) to treat his injury. The ingredients in these medicines are known to damage a betta's labyrinth organ and making it even more difficult for him to breath with a gill injury is not what we want. Do not medicate with anything until we can be sure of how to help you. Randomly tossing in medicine in a panic can do more harm than good. Right now you need to keep your betta as calm and stress free as possible so his condition does not worse. Lot's of clean, warm water and gentle water flow and no dangerous decor or tank mates will achieve this.
First thing is first, what is you Ammonia, Nitrates, Nitrites, Ph, and alkalinity? Since you don't have a test kit you will need to go to a local pet store that offers free water testing. Take a sample of your tank water and have them test it, make sure to get accurate numbers, write them down, and get back to us.
How long have you had the tank?
Is it cycled?
How often do you clean the tank?
Do you siphon the gravel when you clean?
How often and in what quantities do you feed your betta?
Is the flow from your filter strong? Does he struggle to swim near the surface because of it?
Do you have any sharp, rough, or pokey decor in your tank? Shark edges and points are especially dangerous to betta as their delicate fins get caught on things easily. This, combined with a strong current, can cause him to sustain an injury on the decor.
Does he have any tank mates or is he alone?
Have you made any recent changes to the tank like additions of fish, plants, decor, etc?
Can you describe his injury in more detail? It is inflamed or swollen? Is there any discoloration like white tufts (fungus) or pale scales around the wound? If you could provide a close up picture of the injury that would be great.[/ hbubbles come out from his injury...
https://imgur.com/a/E2pnYHx