To strengthen a horse's back, I would suggest you to do a lot of hill work with his head low.
If you can't/In addition of that, you should do a lot of simple exercices (circles, serpetines,..) with the horse on the bit, his back on the fight place and with his hindleges engaged with a good impulsion.
For the galop, it is the same, the aim is to keep the withers up and the hindlegs engaged. To help doing that you can do some pole work.
Good luck, hope this helps :)