I was told the skin can take up to 2 years to heal. I wasn't burned, but I have dermal hyperpigmentation from CO2 laser. I took it that for 2 years, you have to protect yourself from the sun. Most things that correct hyperpigmentation also make your skin sun-sensitive, so yeah, you have to be very diligent about sunscreen while trying to correct your skin.
I've had some success with Microneedling to correct hyperpigmentation, and turmeric is useful as well. Some here have hailed Psoria-Gold, which contains curcumin and is more easily absorbable than a homemade turmeric mask. It's expensive, so I haven't tried it. But over the course of a year I have spent multiple times the cost of Psoria Gold on all sorts of ineffective things. It could be worth a shot.
I recommend wearing a hat, because clothing usually protects your skin even better than sunscreen. Particularly concerning your forehead. But the rest of your face will still need sunscreen. I wore a hat, and no hyperpigmentation formed on my forehead, same can't be said about my cheek. Always wear sunscreen even when it's cloudy and raining.
I was very depressed when I realised that the hyperpigmentation might be permanent. But I haven't given up, and over the course of a year I have seen improvement. It's now been one and a half years after laser. I'm intent on not living with these marks for the rest of my life.
I've been obsessive about healing my skin, which has been important for finding methods that actually work, but it really helps if you can try and go about your life normally. You need to have other things to think about as well. Remember that your identity doesn't depend on your skin. So you need to keep immersed in your passions and hobbies as well. And work, if it is something that inspires you. I find that working can do wonders for maintaining your identity. And being involved in social activities as well, such as volunteering and such.