Europe has some great parks! Though they usually are more family oriented than many parks in the US. So, while many great European parks have a solid one/two coaster punch, the rest of the line-up is usually very family friendly. So, just set your expectations accordingly :).
I think the best region for your trip would be Holland, the west side of Germany, and Belgium. You have some of Europe's best (coaster) parks there such as Phantasialand and Walibi Holland. Plus, the driving distances between parks are considerably shorter than in the US. Amsterdam and Frankfurt also have major airport hubs, so flying should be convenient too. Poland could be an option too (it's quite cheap), but besides 3/4 strong coasters there's currently not much else and there are better parks to visit elsewhere in Europe.
This is by far the most convenient, but traveling by public transport is not impossible. However, considering you only have four days I would recommend renting a car, just for convenience sake. Germany's cheaper when it comes to car rentals so that might also be a factor to consider where you're flying to/from. Lastly, regarding prices (I assume from your post you care about the financials) Europe's overall definitely cheaper than the US (except for gas), including the theme parks so you've got that going for you as well. Hope this info helps and good luck with planning your trip!