Hi @andyroo,
Thank you for your question about addressing a bow in a bookshelf.
While melamine, which is likely what the shelf is made of, is a good material for bookshelves, it is not great at spanning longer distances between supports, and it can sag if weight is applied in the centre where it is unsupported.
Removing the bulk of the weight is a good first step. If you were fine with keeping the weight low and the bow wasn't too frustrating to look at, then I don't think it should matter too much, but if you wanted to increase the weight-bearing capacity and remove the sag, then you would need to support the centre of the shelf.
A good way to do this, although it would block the lovely sign above the window shutters, is to add a piece of timber across the front edge of the shelf to help support the weight applied to it. If you used a piece of dressed pine with a bit of volume, such as this 42 x 42mm DAR Square General Purpose Pine, you could paint it white to match the bookshelf and screw down through the shelf into the top of it and attach it to the side walls with some small angle brackets.
As this is a dressed timber, they are planed down, which makes them nice and straight, so by screwing down through the bookshelf, it will pull up into the shelf, straightening and removing the bow as the screws tighten.
Unfortunately, it would block the sign, but I imagine you could move the Fully Booked lettering onto the front of this support, and it would look just as good.
Let me know what you think and if you have further questions, please don't hesitate to ask.
Jacob
