Ok, so you just moved into your "new" house and learned that the people who owned the house before you added a bathroom on to the children's bedroom themselves and when they put the toilet in they set it too far from the wall so now the toilet tank has no support behind it causing it to flex when someone sits on the toilet bowl.
Sometimes you have to place a support behind the toilet tank to support it if it tends to move when someone sits on the bowl. If the tank moves it is possible that the gasket that goes between the tank and the bowl is too fat preventing the tank from resting on the toilet bowl itself. The toilet tank if installed correctly should sit perfectly on the toilet bowl and not move when it is touched. If it moves too much the bolts that hold the tank to the toilet will cause the tank to crack underneath since the tank was never meant to withstand that type of stress. There are only two bolts holding the tank to the toilet and they were only meant to hold it in place.
If you find that you have a toilet that gets a lot of pushing around, either from someone sitting on it hard or holding onto it for balance you may need to support the tank from behind. You will need to put a board or two by four, whatever takes up the space, between the wall and the toilet tank. That way the tank won't be able to move around and break. It would look better if you could take the tank off completely and install the board but if you cannot do that having the wood show on the sides can be covered up so it doesn't stand out too much.
