There are some good ways to save money and some bad ways. One of the good ways is to avoid some of the commission on materials by providing the more expensive items. It can make a big difference on things like the toilet, towel bars, mirror and vanity light. Sometimes your contractor will also be fine with you providing the tile or flooring, but don’t plan on buying construction basics like plumbers tape or drywall.
Of course, controlling the plan can save a lot. Avoid designs that involve moving the plumbing and especially the toilet, consider things like luxury vinyl flooring instead of heated granite tile, and if you are not moving fixtures you may not need a bathroom designer. If your bathroom feels claustrophobic, instead of expanding the room and shifting fixtures consider a glass shower door to make the room feel larger. Also, once you have a project started try not to change the plan.
Another safe way to save money is by having plenty of time for the project because some contractors will give you a discount for the relaxed schedule. If you can, get the remodel done before you move in or when you can live without the bathroom for a few months.
There are also bad ways to save money because they can cost a lot in less time than you think. These include not getting permits pulled, using unlicensed contractors, or rushing through work before all of the parties involved have been brought up to speed on the plan (for example you may want a look where the glass company needs to work with the tile company).