Those words cannot be used as synonyms.
If you suspect something is true you may believe it is true, but you are not certain. Also, you might not have any evidence to back up your suspicion. If you suppose something is true you are saying that the evidence you have so far leads you to that conclusion. If you assume something to be true you are saying that barring evidence to the contrary that it is true. The word assume expresses the most certainty.
- For Teachers