I think your original idea was right -i.e. raising the bars. It looks to me that you are not comfortable getting down low enough on that set-up, and trying to stay upright which is straining your neck, so you either need to train yourself to keep relaxed in a head-down position, or raise the bars so you're comfortable in a more upright position. Can you flip the stem? Also rotate the bars up a bit? If that helps it might be worth getting a stem with more of an angle - I think the height is the problem rather than length/reach.