Dental clinic

Do you happen to have a HU national helath card?
If you do then you can see a dentist in your district that is assigned to your address. They have dentists that work for national health but they go by st. address as to where you can go. Another idea is to go into your district local center where they have a walk in dental clinic.
You will still have to pay out of pocket for dental supplies if you get a crown etc. Don't forget the tip either, overall it sometimes isn't that much more expensive to see a private dentist.
If you are over age 62 and have national health they let some supply fees slide and give a break on some procedures.
Not sure where exactly you can find the list for your area to go to but your district city hall should be able to give that info.
Perhaps they also have a web site but you would have to read Hungarian to understand it.