How to shift a raster in QGIS?

The Rasmover plugin should do what you want.

You have to allow for experimental plugins to get it in the plugin list.

The result is a virtual raster file, which you can edit with a text editor to adjust the parameters if needed.

I just used this in R and it solved the problem.

Page 195 from https://cran.r-project.org/web/packages/raster/raster.pdf (use table of contents for quick access)

r <- raster()
r <- shift(r, x=1, y=-1)

Edit: Updated the link

For this job I am satisfied with QGIS plugin Freehand Raster Georeferencer. Allows these simple operations in an user friendly GUI: Move, Rotate, Scale, Adjust sides, 2 Points referencer.

Final export consists of three files: TIF, WorldFile (.tfw) and additional information (e.g. projection, etc.) into auxiliary file (.aux.xml).

I usually merge three files into a GeoTIFF e.g. with GDAL. I also strongly recommend to apply some compression method (the result from the Plugin is uncompressed), e.g.

gdal_translate -of GTiff -co COMPRESS=DEFLATE input.tif output.tif