What window insulation can I use for thin gaps?
Air leakage is the primary reason folks apply window film in the winter. By sealing against airflow through the unit as a whole, the volume of air moving through those gaps is dramatically reduced. Obviously this isn't a great solution for the cooling season. 

Otherwise, modern window sashes are sealed either with vinyl tubing or brush seals. You could saw slots into your sashes and install them that way. 

That may not be a practical solution in a rental home. Various types of sash seals are available on the market. Most involve a flange and a vinyl tube that's installed against the sash. Because of your flush frame situation, this could be a bit challenging. Most are designed to fasten to the protruding jamb. I suggest a trip to your home improvement store for a selection.