Is it normal for my skin to swell days after being injected?

Doesn't sound like an infection

I have not read boojum's response yet...that takes a while, but it sounds like a popped vein. That happened to me. When you get an injection or a blood draw, sometimes the tech can accidentally stab through both sides of the vein, which causes bleeding into the surrounding tissue & sometimes swelling and bruising. It's uncomfortable and mildly painful but not serious & should go away after a day or two.

My wife worked on the donation side of the Dutch blood donation service for a couple of years, so she knows a few things about needles in arms.

She says how you explain what happened suggests that you're one of those people who has veins that are very difficult to locate, and your description of what you see on your arm is nothing more serious than bruising. When the needle goes in, it ruptures tiny capillaries between the surface of the skin and the vein, and these ooze blood into the surrounding tissue for a while before the damage is closed with a tiny clot. How long the discolouration last depends on how easily you bruise and how long it takes them to clear.

Hopefully, you won't need an IV again for a long time if ever. But my wife suggests that if you should ever find yourself in such a situation again, you might mention that your veins are challenging, and it would probably be better if they got a very experienced needle-sticker to do the job.

Finally, if you're in a reputable hospital in a developed country and you witnessed the IV needles being removed from their sterile packaging, then it's very unlikely that what you're seeing is signs of an infection.