No higher dimension needed. A space can twist when confined in itself and the twisting can be observed within itself. Alternatively, such change might be invisible (?) to higher dimension. That is, a piece of paper can bend without poking it in the perpendicular direction or trying to twist it around, and things on the paper(of course we know tiny 2d beings don't live on the paper, or do we?) knows when the paper is twisting in on itself, but may not know we are poking it. So essentially a fifth dimension is by no means necessary for spacetime to bend. But that does not discount imaginary time or space. Those who say gravity is an illusion have not felt those beyond what earth can offer.