I had it happen when I bought my home in 2018. The property tax estimates on real estate websites put it at around $4700-4800 per year. So, imagine my shock when I bought it, and initially the property taxes were only around $3300 per year!
When the new tax bill came out for September 2019, it was back up to around $4700-4800 per year. I called to find out what happened. Turns out, that since I had bought the house in September of 2018, they accidentally gave me the previous owner's Homestead tax credit, which over time slows the rate that your taxes go up if the property is your primary residence. But, when a new owner buys the place, the credit goes away, and the taxes reset to the higher level.
It really wasn't that much of a shock to me, as the $3300 seemed too good to be true. But, I could see more extreme examples being a shock. Back in 2017, I had looked at a house, and the real estate listing said its taxes were only about a few hundred bucks per year! This was a $500K+ house. I found out that the person selling was a disabled veteran, or some similar category, where they got a huge break on their property taxes, and online, the real estate listing was showing that number, rather than what the taxes would be without the discount.
Just out of curiosity, I looked the house up online. Apparently it wasn't such a hot item, as it didn't sell until April of 2020, almost three years after I looked at it. And it only sold for $450K, whereas it was something like $533K or so when I saw it back in 2017.
The owner's tax bill was only $383, for years, but then for some reason for 2020 (due Sept 2019) jumped to $2291. And, the latest bill, for 2021, is up to $5547.
I guess I dodged a bullet when it came to that house. I remember it fairly well. It had a big master suite on the first floor, which is something I was looking for. And it had 3 more bedrooms upstairs, a 3-car garage, and was on 2.5 acres. But, it also had no basement, and looked like it was built on wetlands. Most of the property was wooded, and had that look to it like it would flood during a heavy rain. The area around the house and driveway was raised up, but it just seemed like an area that would stay damp. And, if I needed more area, such as extra driveway, more garage space, etc, it would probably be next to impossible to get permitting, because of the low-lying area.
I had to use the map on the assessors web page to locate the property as it didn't go directly with the address as you found. I'm not sure where I got 18.3786 as I also saw 16.3786 on the tax estimator page. Madison Co property tax page is very useful. This isn't the right property for us, way to large, too far from a beach and more than what we would like to pay, but love the land and privacy when compared to a subdivision or in-town. This look into estimating tax on a property purchase for us helps in our continued search. 
