Turbid (whose corresponding nouns are turbidity and turbidness, though the former is favored) means having sediment or foreign particles stirred up or suspended; muddy. It usually applies to water, but it’s also used metaphorically.
Turgid (whose corresponding noun is turgidity) means swollen and distended or bloated, but it’s more often used in a metaphorical sense to mean excessively ornate or complex in style or language. For example, when someone calls a piece of writing turgid, they mean it’s bloated or pompous in tone. Turgid also frequently appears in sports writing to denounce slow and uninspired play, but torpid would usually make more sense in these cases.
Torpid (whose corresponding noun is torpor) means dormant, benumbed, sluggish, or lethargic.
Examples
Turbid
This creates a turbid plume up to two metres deep that shoots across the top of the harbour, the brown sludge that can sometimes be seen off the bow of a ferry on Sydney Harbour. [Sydney Morning Herald]
After reported turbid relations with his father, Muammar al-Qadhafi, in the late 1990s, Muatassim returned from several years in Egypt in 2006 to serve as Libya’s National Security Advisor. [Wikileaks via Telegraph]
Turgid
Goldman’s report was a triumph of form over content, 63 pages of turgid prose that came much closer to inducing somnolence than exhilaration. [Reuters]
Franck’s symphony fell out of fashion decades ago, and my dim memory of the piece was of something grandiose and turgid. [Telegraph]
Torpid
If you’re afraid of ageing and dying (as she was, as I am), then torpid small-town life in the heat is the best solution. [The Guardian]
Throughout the match, he appeared weary, torpid, almost seasick, but often does. [Sydney Morning Herald]