Yep. VSWR most likely is what killed your radio. At 4 watts or even 20 watts, a bad VSWR isn't going to be a big issue. Not add a couple hundred watts to that and things need to be almost perfect to operate correctly. Also make sure you have both batteries tied together in parallel. You can run a POS wire from the main battery to the secondary battery and then the NEG post of the secondary battery to the frame. This goes for the atv battery as well.
But like kf4eyr stated, you need to make sure your VSWR (SWR), is good. Meaning it should be at no more than 1.3:1 with the amp running. Anymore and things start creating heat and also like you see, will burn up radios and possibly the amp as well. You most likely just don't have a good ground plane under the antenna. You need metal under an antenna for them to work correctly. Even the no ground type antennas need some form of ground which is usually the coax outer shield. This would make a lot of sense as to why your radio blew. As your coax becomes the ground wire that connects to the radio from the antenna. If you are using a no ground plane type antenna like a firestick or similar.
You first need to tune the antenna without the amp in line. Then add the amp. Which on an ATV I just wouldn't suggest unless you have a good grasp on RF energy and how it works and also the other things that are involved with a properly tuned antenna besides a decent VSWR. There is more to tuning antennas than just the SWR. For now I would pull the amp put of line and just use a radio. I just don't think you will get a good enough match to allow an amp to be added to the ATV. JMHO's. Also you may have created a ground loop by not having both batteries going to the some ground. As in if the ATV battery is grounded to the frame, so should the secondary battery. The way your photo shows, both batteries are completely separate units. And like said, if you are using a NGP type antenna, you are just peeing in the wind. Read up on how to properly install 2 batteries into a vehicle or an ATV, and also how the no ground plane antennas actually work. If that is what you are using. If you have a 102" whip and your ATV body is plastic, I can about guarantee you didn't have a good VSWR or enough metal under the antenna to create a proper match for the antenna. Again, these are JMHO's. But I am pretty sure they are correct. Like I said, time to start reading about some things if it were me, that is what I would do. And check the ATV forums and see if anyone has added an amplifier to a CB radio on an ATV and how they did it and how well it worked. Lots of info out there. Just have to take the time to find it and read up on it. Hope you get your issues resolved, but IMO, I would not run an amp on an ATV unless I knew 110% for sure that I had plenty of metal under my antenna and also a good match. This will require more than just tuning for lowest SWR, you will need someone with an antenna analyzer to help you with this to do it properly. Best solution for now is just to leave the amp for something else and either get your radio repaired or get a new one. You didn't mention what type radio you have, but if it's not an expensive one, I wouldn't even worry about getting it fixed as it may end up costing you more than what you can buy a new radio for. Again. JMHO's. Hope this helps and hope you get your issue resolved. Have a good day and 73 and God Bless.