Is it normal to have to work when injured?

If you have a punctured lung, you probably shouldn't be working.

Don’t be a people pleaser, your health should always come first

A punctured lung? That's far from a bruise or sprain.

That guilt trip move some bosses pull is such bullshit. Don't ever put your health at risk over a job. And anyway, I'd think that because of your solid work ethic your boss wouldn't have a problem giving you some time off for health reasons. No one can fire you for being sick, no matter how low on staff they are. You could sue and win.

Everything said so far is true.

My advice is find out what the sick policy is, like how many day or hours of sick time you have. Then you will know how much time you can take off without risking your job. If you can get to a doctor, he may say you need more time off than whatever you already have.
There may be some paper work involved if taking extended leave.

If one is planning only a day or two to call in sick, I would say call and just say, "This is so and so, and I will not be making it in today". The reason doesn't really matter. If they ask, say, "Well something came up..."

For the guilt thing - As already said, that is not your problem if they are understaffed.

Bosses are often trying to find out who they can or cannot walk on. I think the ones at my place of work see me and are like, "Oh damn, just Brenda, she ain't about to out up with it."

It's not your fault it's understaffed. That shit is their problem to deal with, not yours. Get a doctor's note, show them your condition is real, and relax at home for a few days.

Its called standing up for yourself. You didn't do it and you should have, and now you'll regret it.

Just say you can't go to work. Stop feeling guilty about shit as well, no one else hardly does so you're better off not caring.

Tell them you can't work and bring documentation if your employer requires it.

Employers will work you to death if you let them, it's up to you to stand up for yourself.