Each item is a potential problem that typically drives up costs. Following each item is a suggested solution.
1. Size of the bathroom will change
Solution: Resize only if absolutely necessary to accommodate your needs. This is the single most expensive aspect of the bathroom renovation; avoid at all costs.
The reason is because plumbing--particularly the toilet discharge and sewer pipe--are expensive to move.
2. Load-bearing walls must be removed or moved
Solution: Explore possibilities of expanding through non load-bearing walls--walls that do not bear weight. These walls can easily be removed using simple hand tools. Often, no permits are required for this. As a rule of thumb, exterior walls tend to be load bearing. Interior walls that run parallel to ceiling joists tend to be non load-bearing.
If you do want to move that load-bearing wall, it is possible to do on your own. Materials do not cost a lot. Mainly, you need to correctly calculate the load and purchase a beam suitable for carrying the load.
3. Walls cannot support additions of new vents, ducts, wiring, windows
Solution: Walls do not necessarily have to be totally replaced to accept these additions. Bad studs can be sistered to increase their load-bearing capacity.
Raise this issue again with your contractor or seek a second opinion.
4. Drywall is water-damaged; needs full replacement
Solution: Drywall often must be completely replaced in bathroom renovations due to the high moisture content; this is common enough and should be anticipated. Confirm with your contractor that full replacement is needed.
It's possible that only the affected areas need to be replaced.
5. Sink, tub, and shower fixtures to be removed and replaced with new ones
Solution: Do they functionally need to be replaced or just aesthetically? They may be ugly but operable, and can be replaced easily by yourself at a later time. If that fails, buy on your own and supply to contractor. Or instead of replacing, you can install a liner over your tub or shower. If you don't like the idea of a liner, you can always refinish your bathtub.
6. Old shower to be removed and replaced with expensive tiled shower
Solution: Consider a pre-formed, one-piece shower stall rather than a tiled shower. It will be considerably less expensive because you won't be hiring costly tilesetters. Also, pre-formed shower stalls go up in hours, whereas tiled showers take several days.
7. Toilet outdated or functions poorly; replace at high cost
Solution: Neither materials nor labor should drive up this cost. Toilets should not have to be expensive to do the job right. Toilet installation is a common do-it-yourself project, so the contractor should not be charging you high labor costs. In fact, this is a project that homeowners routinely do by themselves.
8. Outdated electrical system must be completely rewired
Solution: Why? Because your system does not meet the current electrical code? If it is functional, your local permitting office may be able to tell you if your system can be grandfathered in. If it needs to be brought up to code, it may just need tweaking.
One common example is to replace non-grounded outlets with GFCI (ground-fault current interrupter) outlets. Have an electrician who is not associated with the contractor give you a second opinion.
9. Completely re-plumb bathroom because sink, tub, and shower are being moved
Solution: Reconsider plans you have for moving these items. Moving plumbing is second only to bumping out walls when it comes to incurring costs. When you consider the plumber's high estimates, you may find that you can live with your current layout after all.
10. Plumber quotes excessive price for bathroom plumbing work
Solution: If cost is the sticking point, consider taking on your own plumbing work. With the advent of PEX pipes and Sharkbite fittings, homeowners can now DIY their own plumbing work for a fraction of the cost of a plumber.