It looks like that Caroline married in the USA, judging from Caroline age on immigration and no record over in England.
From the info you provided - this looks to be her in England:
Piece:546 Book/Folio: 3/9 Page:12 Oak Street,Oldham Above Town, Lancashire.
KERSHAW, Samuel M 39 1802 Cotton Weaver, Lancashire KERSHAW, Martha F 38 1803Lancashire KERSHAW, William M 18 1823Lancashire KERSHAW, Robert M 13 1828Lancashire KERSHAW, Adam M 10 1831Lancashire KERSHAW, Violet F 5 1836Lancashire KERSHAW, Caroline F 4 1837Lancashire KERSHAW, Emma F 0 (1 months) 1841Lancashire
Her parents marriage:
Marriage: 22 Jul 1821 St Mary, Oldham, Lancashire, England Samuel Kershaw - (X), Weaver of Edge Lane Martha Robinson - (X), Spinster of Edge Lane Witness: Ralph Jackson; Jonathan Jackson Married by Banns with consent of Parties by: J. Fallowfield Register: Marriages 1818 - 1822, Page 239, Entry 715 Source: LDS Film 1656118