Julie,
No wonder you haven't been able to find them on the 1871 census. I've found them and the handwriting on the census folio is very poor.
The index has the family down as "WEARTURCH"
Source Citation: Class: RG10; Piece: 4934; Folio: 115; Page: 40; GSU roll: 848020.
Address: This is very bad hand writing. It looks like Beechburn Mill? Newton Cap, Durham. This is within Registration district: Auckland
Sub registration district: Bishop Auckland
Recorded as follows:
John Wearturch Aged 37 Head Married Joiner Born Gar(i)nford Yorkshire
Jane Wearturch Aged 32 Wife Married Born Walshingham Yorkshire
Ann Wearturch Aged 5 Daug UnM Born Witton Le Wear Yorkshire
Sarah Wearturch Aged 3 Daug UnM Born Witton Le Wear Yorkshire
Jane Wearturch Aged 2 Daug UnM Born Witton Le Wear
Yorkshire
John N Wearturch Aged 5 Mths Son UnM Born Newton Cap Yorkshire
The actual county given is done so via a Ditto carried down half way up the page from a birth place of another family. As I'm not familiar with the area, I'm sure that the county is recorded incorrectly due to another mention of Newton Cap on the page as being in Durham. I'm sure someone from around the area can confirm this. John and Jane's birth places are so difficult to read, these are my own transcriptions.
Any transcriptions in this post are Crown Copyright,from
www.nationalarchives.gov.uk