Green’s Dictionary of Slang

(US/UK) slim, skinny; thus narrow-minded.

1833		Satirist (London) 12 May 6/2: Sir,—Dont you think its a most abominable shame that a set a narrow gutted varmint like you should go to come here for the purpose of extirpating a profession which possesses the honour of having me for one of its most distinguished members?
1916		H.N. Cary Sl. of Venery.
1953		J. Franklyn Cockney 275: A thin man whose nickname is actually ‘Narrowgut’ will be known simply as ‘Narrow’.
1965		H. Rhodes Chosen Few (1966) 136: His narrow-ass deputy and a coupla white MPs are on their way over.
1974		D. Ireland Burn 109: Not if my glorious narrow-gutted, narrow-minded Australian workmates had any say in it.