turn down vs lessen

turn down and lessen both are verbs.

Word	Noun	Adjective	Verb	Adverb
turn down	No	No	Yes	No
lessen	No	No	Yes	No

As verbs, lessen is a hypernym of turn down; that is, lessen is a word with a broader meaning than turn down:

Other hypernyms of turn down include decrease, minify.

turn down (verb)	lessen (verb)
make lower or quieter	decrease in size, extent, or range
take a downward direction	wear off or die down
reject with contempt	make become smaller
not accept as true
refuse entrance or membership