Some thoughts-
Property tax is almost always charged as a percentage of your home's assessed value, so even within one city/town if you buy a home that's double the value of current your home then you'll likely that your property tax will be much higher too. (Probably double.)
You also mentioned that you're moving to a suburb. I don't know of you currently live in a more urban area but that could be a factor too. From the little I've seen, I think the less urban you go the higher the property tax is. Think about how much road, sewer, snowplowing etc there is per house in a dense city vs a more rural area. Things like police, fire and schools may also be much more efficient to operate (cheaper) in a city.
Many jurisdictions do have a system where you can appeal the assessed value of your home to try to lower your taxes, and there are consultants/lawyers that can help you with that. However, you gotta figure out if that's worthwhile in your case. If someone has a 30k property tax bill then it'll obviously make sense for them to pay someone to appeal ot and try to lower it 10 or even 20%, but with a $4k bill you gotta figure out if it's worth paying someone to fight it. Also, see how long your assessed value will remain in place if you win an appeal- don't pay someone a ton to appeal it if it'll be reassessed next year. Maybe in some places the appeak process is simple enough to DIY- I don't know.
Also before appealing try to figure our if you have a case at all. In some places that I know of most peoperties' assessed value is way below market to begin with (as they only reassess every few years and the market has climbed quickly) so there's almost no way you'd win an appeal if the assessed value is already low. If you think you actually have a case (i.e. you think the assessed value is higher than what it's worth) then maybe try to fight it.
Oh, one more idea- figure out how to transfer your house to a non-profit to waive taxes completely in many jurisdictions. /s