We have removed the previous flooring in our second floor bathroom (tile over vinyl) in plans of placing porcelain tile flooring. The subfloor is plywood but it is in bad shape (water damage)and we plan to replace most of it. The damage was not noticeable because the previous owner had covered it with tile flooring but not repaired the real problem. The floor joists are 2 by 10s, 16" on center. The unsupported length is 13'. My concern with replacing the plywood is that on both sides of the bathroom the floor joist is about 4" from the wall. If I cut the plywood at the wall then the two walls will be floating on plywood that is supported over one floor joist. Do I need to restructure the framing for the flooring to support the walls in these areas? This will be a problem since it is on the second floor and we have just remodeled the family room ceiling on the first floor.

A four inch gap is not really to important but to do the job correctly Yes you must support the edges of the plywood on all sides. You simply need to pack out the edges. Not so much that it will drop along the edge, but it will lift as you place pressure in the middle of the joist space next to it. It may seem like its not moving, but that little amount will be just enough to crack the new tile and cause the grout to fail.

To do this you need a drill, some three inch screws, some bar soap to lube the threads of the screws. To start off you need to pre-drill some pilot holes in the joist that is running along the wall. Cut a 2x4 to fit into the space. While someone holds it into place use the drill and some three inch screws to fasten the board into place. Thats all there is to it!
It would not hurt to use a little liquid nails between the new and old boards as well.
Be sure to cross block where the plywood joint will be as well.

I would also place blocking between each joist along the ends to prevent any bowing that may occur in these areas. Not diffcult but will take some time. It always does doing it right.