so-called vs supposed

so-called and supposed both are adjectives.

Word	Noun	Adjective	Verb	Adverb
so-called	No	Yes	No	No
supposed	No	Yes	No	No

As adjectives, so-called and supposed are synonyms defined as:

Other synonyms of so-called include alleged.

so-called (adjective)	supposed (adjective)
doubtful or suspect	required or under orders
	doubtful or suspect
	based primarily on surmise rather than adequate evidence
	mistakenly believed