Trainer bike tire does not go on nicely, am I doing something wrong?
Make sure you are fitting the tire correctly. Examine the bread all the way around the rim on both sides and make sure it is seated evenly.

I wonder if the inner tube is not properly inside the tire near the valve. The reinforced area around the valve can get stuck under the tire bead. Before inflating, push the valve into the rim to make sure the tube is properly inside the tire.

If you still have problems try marking the tire where it popped off the rim and mounting the it rotated 180 degrees around the rim. If it comes off the bead in the same place the issue may be damage to you rim that the loose fitting tire is revealing. If the same section of tire pops off the tire bead may be damaged at that point.