Your dog is walking in herbicide - you have applied weed killer to an area used by a pet.

https://www.scotts.com/en-us/products/lawn-food/scotts-turf-builder-triple-action

Found this http://gooddoggies.online/dog-pee-killing-grass-3-fixes-for-your-lawn/

the problem might have been compounded because you just fertilized. Training the dog to go in one spot away from your house is nice. Walk them out on a leash until they know that is an approved spot. They will eventually just go there.

Dilution is the key. Urine is extremely nitrogen rich and the pee is burning the grass just as if you had overfertilized. Which the dogs did for you :-)

And the above comment is very valid with pets or even small kids. Granular products are easily take up by furry pads or bare feet/hands and too easily transferred to the mouth. Both pets and kids have a much lower tolerance to any sort of pesticide and I would lean very heavily towards organics in your case.

This info says supplements that balance the urine ph don't work. It really is the added nitrogen http://thebark.com/content/keep-your-lawn-free-urine-spots

Thanks everyone. This is very unfortunate as it is somewhat unrealistic to dilute the area every occasion. I don't really have a designated area that would be better for the dogs, so I will have to come up with something else.

designated really is just someplace away from the area of the yard you use. farthest corner from the house. It really isn't that hard. I got a puppy once New Year's Day. I thought it was great he went into the corners on his own. What was really happening was he was "going" where the snow was left in the shaded corners of the yard. He was following the snow. He thought that was where he was supposed to go. Crazy. I Figured it out when there was no snow left In the yard.

Lymphoma in dogs has been linked to lawn chemicals. A couple that lived near me used to walk their two lookalike (same breed) dogs off leash in my area. One time when I encountered them with my leashed dog there was a graveled street-front section of property (border or parking strip) right where we were that positively reeked of a herbicide application. (As in a soil sterilant type). No chemical application warning signs were present. I pulled my dog away, they paid no attention regarding theirs.

A few years later I found out both of their dogs had gotten cancers and died of them.

After the fact, periodically spread garden Lime to neutralize the Ph.

" Concentrated urine has more solutes (particles which can affect grass health) than dilute urine. Female dogs’ urine may be more likely to cause more grass damage than the urine of male dogs because females typically squat and urinate in one spot, whereas males tend to urinate in smaller amounts as they wander from spot to spot or spray urine over a larger surface area. Making sure that your dog is well hydrated can help maintain a less concentrated urine, but this will not necessarily prevent the problem entirely.

Dogs typically have a fairly neutral urine pH, between 7 and 7.5. Diets that are very high in protein content will often lead to a lower (more acidic) urine pH. "

Canidae

Dog urine is essentially pH neutral. Applying lime will have no effect on the spots. That is due entirely to nitrogen burn.

"The Three Reasons a Dog's Urine Burns the Grass

There are three primary reasons why dog urine burns grass: alkaline
urine pH, the concentration of the urine, and its nitrogen load. The
most important of these factors is urine pH. The best way to find out
which is the causative factor in your dog's situation is to drop a
urine sample off at your vet for a urinalysis"

By Dr. Becker

If you look at the picture at the top of the thread showing the dead spots, you will see the effect of the high nitrogen burn - dead in the middle where the pee was most concentrated, surrounded by an area of lush, very green growth as the concentration drops as the pee spreads out. So the key is to dilute the concentration of the central spot ASAP! A neighbour with a large female Rottweiler and a perfect lawn provided the ‘lightbulb moment’ for me many years ago when I saw him grab a filled watering can to immediately dilute the pee when his dog squatted.... He kept a couple of filled watering cans in key spots around the yard so there was always one nearby when the dog peed. He was always with the dog whenever she was out - i.e. she was never left in the yard on her own so he was always able to dilute the pee as soon as it happened.