This commonly happens when working with raster basemaps, especially if they come from online sources.
Your polygon didn't shift, it was the raster behind it that shifted. Rasters and on-the-fly projections are not good friends (at least with the technology we use nowadays, hopefully this will change in the future).
If you want the raster to stay in place, you need to make sure that the raster source is in the same projection as the current project. This could mean you need to make a local copy of the Bing map (check if you the licence allows you to do it first) and reproject it (using the geoprocessing tools) to the current projection you are working in.
Then it shouldn't move.