In fifth edition magic users can not cast healing spells - that is the role of the cleric. Paladins and druids can do so as well at a lower level.
Also, fighters in 5th have an ability called "second wind" that can help a lot. And, as someone already mentioned, if healing potions are moderately available (in dungeons or for sale) they can help solve the problem - and provide a way for the DM to siphon gold off of the party.
I will hopefully get to work on the character tomorrow - maybe a cleric. Or maybe not.