suppose is like assume and you use it when you aren't sure about something suspect it's used when we think that someone made a crime but we aren't sure this man is suspected of killing the woman

Suppose is when you have evidence for your thought. He punched a kitten, so I suppose he punches puppies too. Suspect is when you don't have evidence (or don't have very good evidence) for your thought, but believe it anyways. I suspect he punches puppies, but I have no way of knowing that for sure.

Both words are synonyms. Both words mean "guess", however suppose is used more often when we really don't have any real information on which to base our supposition (our guess/opinion). "suspect" is more often used when we have some reason to believe that something is true or untrue, and suspect is often used when talking about a crime or an action that caused harm or damage.