Up the creek
There is no doubt that up shit creek, and its more euphemistic version, up the creek, are of American origin from the 19th century. In North America, a creek signified a narrow, secluded, difficult waterway, hence up the creek meant being in trouble or difficulties. During the early 20th century, the phrase ‘without a paddle’ was suffixed to up the creek as an intensifier, to indicate a situation of even more difficulty. The OED dates the appearance of these latter expressions on the British side of the Atlantic as 1937 and 1941 respectively. The origin of this phrase has nothing whatsoever to do with the Royal Naval Hospital in Gosport founded in 1753, which is situated on Haslar Creek. For many years, the Royal Navy’s sick and wounded were treated there until its closure in 2009. Such patients may well have been ‘up the creek’ but the alleged etymology is purely apocryphal.