"We are running towards the hospital."
Translation:Hastaneye doğru koşuyoruz.
'-e doğru' means 'towards'. Without '-e doğru' it would be 'We are running to the hospital.', which is not the same as running towards the hospital. When you run towards the hospital, that does not necessarily mean that your target destination is the hospital. It just means that you are running in the direction of the hospital. Maybe you are just jogging and your route is in that direction, for instance.
GordonRobb, have a look here, about propositions and posposition in English grammar, very clear: https://en.wikipedia.org/wiki/Preposition_and_postposition#Classification_by_position