To swear means to promise or pledge. We often swear on something, like our honor or our life, or to a deity.
Ex: I promised I would return, but she was still worried. I could tell that she didn't believe me, so I swore on my honor that I would come back tomorrow.
Ex: Sarah insists that this is the best restaurant in town. She swears that only there can you taste the best pepper steak of your life.
To swear can also mean to use profane and rude language. When someone swears a lot or with particularly offensive words, you can say they swear like a sailor.
Ex: You cannot swear on public television. It may upset people and more importantly, it violates broadcasting communications standards, which forbid the usage of profane language.
Ex: You need to teach that girl some manners! She got carried away playing video games today and started swearing like a sailor! I couldn't bear to hear such language!
Assure means to tell someone in a very strong way that something will happen or is true. We also use assure when we make something certain or true. In this case, a synonymous verb could be the verb ensure.
Ex: I know your leg hurts, but I can assure you that Sarah knows how to help you. She is an Accident and Emergency doctor.
Ex: You must not move your leg at all to ensure that it heals properly. If you move it, the bones will not heal.
We commonly use the phrase rest assured when promising or telling someone that something will happen. It means you can be sure, or I swear this is true.
Ex: After Salina's purse was stolen at the library, the security guard told her not to worry, but to rest assured that they would find it soon.
Ex: Sam is working very hard today, so I am certain that he will not be late. We can rest assured that he will finish the report by tomorrow morning.