Prepare the substrate properly. That means ripping it all down and adjusting the walls and backerboard with shims until it's all level and plumb. It's currently NOT level and plumb.

A picture would help.

What Hollysprings is describing would be a correction of a bow in the wall that your tiles are traveling on. If it were cupped, you'd have a squeezing of the grout spaces. To illustrate this take a fabric tape measure and measure the inside of a coffee cup, then measure the out side. It will be a longer measure on the outside, right? That is what is happening to your walls.

I'm guessing you're doing one row at a time, and not stair-stepping the tiles as you install them. You probably are just not maintaining a true plumb (or level) installation. Or the old tile was not plumb. It's not clear to me if you're tiling up the wall (in rows), or across the wall (in columns). A picture would help. Since it sounds like the only grout joint that is widening is the one between old and new tiles, I don't think you have a bowed wall problem.

Catbuilder, I re-read the OP and you might be right, either the original isn't plumb or the new isn't plumb.

Here you can see the space on the right side widening as it goes up. You can also see how I tried to keep the vertical lines lined up between the tiles as I went up the wall.

From what I can tell, it honestly looks like they had originally done the tile on drywall.

This is what is behind the older tiles.

Thanks for the pictures. Big help.

Is the row of tile going up the wall by the window plumb and is the bottom row level?

I suspect one is out of whack and when you set the new tiles perfectly plumb, it highlighted the problem.

It's hard to tell by the picture but there looks to be a big gap in the corner which could be a caulking issue down the road. I would have started on the right edge with 1/2 or 3/4 of a tile so there would be a tighter fit in the left corner.

IMO, if you aren't prepared to rip the wall down to studs and start over, finish tiling and live with the fat grout line. As you go up, nudge the tiles over to close the gap.

If you're talking about the left side where there is a column of tiles missing, it's because those would need to be cut to fit in that space.

I don't know if the row by the window and the bottom is level, I just used those as my guide assuming they were level and I would have no problems just sticking the same size tiles back in there.

I haven't worked on it in probably a year or two now, just because I got annoyed with the whole thing, plus we have the guest bathroom we can use instead.

The pictures are even more illustrative of the fact that the entire installation needs to be removed down to the studs and started over. It's NOT waterproof, and other than gaining some practice tiling, you're essentially wasting your time here. It's got to be done correctly from the studs out.

If it's tiled directly on drywall, it is not waterproof. You should start over and waterproof it. It also looks like mastic was used and not thinset. You should NEVER use mastic in a shower, even if the manufacturer says it's ok. Mastic will re-emulsify when it gets wet, which it will. By the way, pre-mixed thinset is not actually thinset, it's mastic.

The new stuff is on hardie board or cement board. I also don't know how you can tell what I used, but yeah I'm pretty sure its the pre mixed stuff.

Is there anything BEHIND the backer board? You need a moisture barrier either behind the backer board, or on the front of it. You're "pretty sure" it's the pre-mixed stuff? If you didn't mix it yourself, then it's pre-mixed.

Yes, there is a barrier behind the backer board. And yes it's the pre-mixed stuff. Sorry I haven't worked on this project for approx 2 years now, so I had forgotten if I had used the pre-mixed or actually mixed the powder stuff together because I actually had both at one point.