Sorry for the troubles. Sounds like the ladies had a few spats and lots of stress.
I know how you feel and have been ready to chuck it once or twice myself.
I am inclined to suggest that you get a few critter keepers and isolate the worst girls out to treat.
Rereading your post, it sounds like you have 4 girls in the tank. Adding new girls to a tank is a complex problem, and 4 girls are in most cases too few. 5 or 6 is the minimum recommendation. Separate the girls; take the sick girls out. Do frequent water changes and get the water tested. Get a test kit if you can. I think your tank cycled, and you have an ammonia/nitrite build up. If so, change the water often--at least daily--until it tests right
I have ADF's, plants (though young), and Botia straita (loaches) in my sorority tank. I frequently add some aquarium salt. They all do fine. I have one girl in the sorority who has some kind of "something" on her head that gets bad without salt. Her sister died with it, but she maintains as long as I keep some salt in her water. I don't add lots of salt because of the other inhabitants. It is a twenty-five usg tank. I add 1 tsp or 2 to a cup of hot water, stir it up and dribble the salt that disolves into the tank. If I don't see improvement I add a bit more the next day or so. I know, it is very "wing it," but that's what I do to add salt. The ususal recommendation is about 1tsp to 2.5 usg. I never use that much unless I am doing a dip or bath or an isolated sick fish. As you probably know, salt will stay in the tank until water changes and only diminish by the % of water changed. But it will diminish each time the water is changed, so you can use it for immediate treatment and is an excellent combatant of Betta fungal infections.
My guess is that there is a fungus and a bacteria in the tank. I use Maroxy for fungal infections and salt. I like Mardel Products because they do not kill the good bacteria or require the removal of carbon from the filter ususally. These are not available in the UK.
Keep the water very clean and change it often. A mini cycle may have started, when the new fish were added.
I will let others address the bacterial infection if any. The girl on her side is in serious trouble and may not recover. I would most definately remove her from the tank in any form. The sick fish need to be separated and the water changed very frequently. Therefore they can be in smaller unfiltered containers.
For a new and minor case of ich, I use Aquarisol, which is mild enough to be used with i=other meds. Once the other infections are cleared up, you can switch to a stronger ich med if it has not cleared up . You may need to add an air stone to the tank.
Now to sum up: There are several infections in the tank stemming from several possible events. They are: water quality from the new fish and a mini cycle (this assumes that the tank was already fully cycled); fungal, bacterial, parasite infections resulting from the water quality deterioration and a possible fin nip/damage; stress from the new fish and habitat changes and deteriorating water quality and increasing sickness.
To do: remove the Bettas to separate containers that can have frequent water changes and be treated individually. Do major and frequent water changes in all tanks. Get the water tested and get water test kits. Treat the main tank and all the Bettas with an anti fungal and an anti ich med. (Mardels's CopperSafe can be used safely with Maroxy.) Treat the Bettas with salt. Add a little salt to the main tank to improve gill function with an air stone.
I will not address the possible bacterial issues. If treatment for bacterial infection is needed, Mardel's Maracyn and Maracyn II are a frequent choice. Mardel is one of the most expensive medication choices and is milder that some other medications.
There are different effective treatments. Others may suggest another regime and it may be cheaper and stronger. These are choices you will need to make.

I'm sorry this was so long and complex with some contradictions. I was tinking it out while trying to address and understand your problem. Please believe me when I say, that any experience fish keeper has had these types of problems. Also if I have said to do something you are already doing, just realise that I am starting from the most basic requirements, and am answering w/o much information about your understanding and experience.
You could read the Betta care and Betta health sections at this site
here
Our pinned sections are also good.