All cards are laid face up in 4 rows of 13 and the goal is to get into one pile but you can only move cards that are next to each other or 4 spaces away.
I believe you are talking about Gaps (or some variant of it).
The cards are dealt into four rows of thirteen. The aces are removed and discarded from further play. The gaps that they leave behind are filled by cards that are the same suit and a rank higher than the card on the left of the gap. (For example, 4♣ can be placed beside 3♣.) However, any gap at the right of a King is considered dead and no card can fill it.
Quote from: https://en.m.wikipedia.org/wiki/Gaps
Possibly related to Accordion Patience, where a pile of cards can be moved either on to the previous adjacent pile or the pile that is three before. The aim is to reduce the cards to a single pile.
It differs from your description in that the cards are treated as being in a single line rather than in 4 rows of 13 cards, although for space reasons you could lay out cards in that form. It's not stated in your description how adjacency works for cards in different rows - and given your "next to each other or 4 spaces away" - you might be considering edges of the rows to connected in a manner that's equivalent. Accordian Patience typically uses 3 spaces away rather than 4 but that seems a small change.