I am very familiar with this pain in the bum problem!. What you have is a pretty ideal system for removing plastic plugs from the wall. Plastic plugs work extremely well when the fixed part (bracket) is in close contact with the plug and wall, but when there is space between the plug and the bracket, the screw just pulls the plug from the wall.
For a problem like this, where there is space in front of the plug, I have a couple of solutions I usually try. The first is to drill much deeper, at least twice as deep as the length of the wall plug and use two plugs similar to the ones Doug71 suggests, or a long hammer plug like this. Hammer in most of the way, then final tighten with screwdriver. Sometimes I remove the hammer screw and replace with a suitable long screw. The fischer wall plugs you are using are fine in a situation where the centre section can compress and expand, but I find they often do exactly what you describe.
If you use two standard length plugs in the hole, the first can be knocked in using a long screw, Partly screw into the plug, then knock to the bottom of the hole. What you are trying to achieve is having the screw well into the wedge section of the plug before it starts pulling the bracket to the wall, so the screw is anchored well I often use a touch of oil on the screw to reduce the torque friction. That can help make them work better.
For a problem like this, where there is space in front of the plug, I have a couple of solutions I usually try. The first is to drill much deeper, at least twice as deep as the length of the wall plug and use two plugs similar to the ones Doug71 suggests, or a long hammer plug like this. Hammer in most of the way, then final tighten with screwdriver. Sometimes I remove the hammer screw and replace with a suitable long screw. The fischer wall plugs you are using are fine in a situation where the centre section can compress and expand, but I find they often do exactly what you describe.
If you use two standard length plugs in the hole, the first can be knocked in using a long screw, Partly screw into the plug, then knock to the bottom of the hole. What you are trying to achieve is having the screw well into the wedge section of the plug before it starts pulling the bracket to the wall, so the screw is anchored well I often use a touch of oil on the screw to reduce the torque friction. That can help make them work better.