I got an iPhone 13 Pro a few months back and was shocked at the photos too. They all looked over sharpened and over saturated, with too much contrast. Everything looked like bad HDR, but unlike my previous phone (iPhone XS), my new phone wasn't saving two photos, and there was no toggle in the settings to turn off HDR. I seriously considered returning the phone, but my XS was all out of space, so I really needed the upgrade.
But I fixed it!
What worked for me was doing as follows. Settings > Photos > HDR (HIGH DYNAMIC RANGE) > View Full HDR > Off. It's a toggle that's on by default. Once I turned it off, my photos all looked much better, as I was used to.
Apparently, for the iPhone 13 series of phones, Apple doesn't let you choose to turn on/off HDR in the camera settings, and it doesn't display two images in your photo gallery. Instead, your phone will always take both, and you choose whether it displays the standard or the HDR version from the photos settings.
This applies to any photo you've taken with your iPhone 13 since you started using it. All the photos I took - which I thought turned out horrible and couldn't save - suddenly looked great.
It's still kinda troublesome if you might want HDR for some shots, but at least you have both.
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I do have a problem with the 3x photos, though. The iPhone 13 Pro will zoom, but it decides on its own whether it's using the telephoto lens or digitally cropping in on a wide photo.
The cropped images look pixelated and over sharpened, but it's difficult to tell which lens is being used from the live preview. The photo does state the camera used in the EXIF data, but it's usually clear enough from the image itself.
It's possible to frame things such that it'll trigger the telephoto camera, but it's very fickle and can snap back anytime. It's annoying, and I wish they'd add a toggle like they (kind of) did for macro photos.