With the right cards this can be fairly easy. In this challenge you acquire curses as you progress from collecting the cursed objects required to close the portals. The third cursed object gives you a curse that will generate further curses.
Within your encounters deck you will want The Demon Arae, The Healer, Cunning Man, and Forgotten Dreams. You can also bring The Little Devil, although this card will actually end up giving you a curse if you have no curses to give.
The Demon Arae will remove up to 3 curses, based on your dice roll. The healer can remove as many curses as you want as long as you have fame. The first curse costs 8 fame to remove, second costs 10, third costs 12, and so on. The Cunning Man will accept a curse as a trade, and forgotten dreams lets you replace any of these you have already encountered.
The equipment you will want to bring is actually what can make this exceptionally easy. Spiked Boneguard will prevent the next curse you would receive, but the biggest boon here is King's Crown, which lets you sacrifice your companion for a few turns to prevent a curse. If you can get the King's Crown within the first 2 maps you should have the fame to use it by the third map, which will then allow you to prevent all further curses you would gain with it's effect, thus diminishing the amount of curses you need to actively remove to 2, which should be easily accomplished with the curse removal cards in the deck.