We don't really subsidize fossil fuels. I thought that your comment was interesting, so I looked it up. These "subsidies" are just completely normal stuff that ordinary American businesses get.
Mostly it is tax deductions. Everybody is getting tax deductions. Everybody from your local hair salon to Microsoft is taking tax deductions. For the fuel companies this includes income tax, fees for shipping, and royalties for extracting the resources. The only really offensive one is a deduction for BP's punishment, which sort of undoes the punishment... though that is fair if the tobacco companies got to do likewise for their punishment.
The rest is just the unpaid share of the cost of running various government agencies. First of all this isn't generally something the companies feel they benefit from; ditching OSHA/EPA/etc. would probably please them. Second of all, again it is something we do for all American businesses.
As with any other American businesses, wind and solar providers get the subsidies. They also get special environmental subsidies, which are huge.
I think this dispute started because the abnormal subsidies being provided to wind/solar providers have become a political issue. The wind/solar providers respond by pushing the narrative that fossil fuels get subsidies, but that just isn't a reasonable conclusion. The supposed subsidies are just normal things provided to American businesses, unlike what wind and solar are getting.