It is possible that they moved and may be just a coincidence that the house numbers were the same.
If I were you I would have a look again at the family on each census. If you look at the neighbours, and around the nearby streets, writing names down as you go, you will be following the enumerator's route and get a feel for the layout of the area.
This way you can compare the New Row and West Row locations to see if they are similar or not. You can also check oldmaps.co.uk to try to locate the streets on maps that correspond date wise with the censuses.
A simpler way is to go to the first page of each census where there will be a description of the enumerator's route which may, or should, mention the names of the streets. So if you see both New Row and West Row on the same census you will know that your family moved.
Looking at the names, it seems possible that New Row was a later addition simply due to it's name.
Sorry that's a bit long winded, but hope it helps.
