Well as an Ex plumber there could be two reasons for the splashing the first as described above is simply too much water going into the bowl reset the inlet water valve to a lower level and see if that works or get a bottle an old pint milk one is superb if you have one fill it with water and place in the cistern out of the way of the float and mechanism. This will not only cut down on the amount of water being flushed bit it will save you a lot of water as well. What is in the bottle never gets flushed.
If that does not work then it could be that the sewerage outlet pipe from the bowl has a partial blockage in it like tree roots etc this will tend to hold back the water and make it rise in the pan so it splashes. Get this checked by a qualified plumber and if the old drain is clay type pipes and is cracked or something and it is tree roots, get it all re plumbed in PVC pipe with glued joints. A quick fix and a temporary one is to get the plumber to use his drain cleaning machine on that drain. Cheaper in the long run to get the new pipes put in though.
If you have a long flush pipe on the cistern to the toilet bowl you might need to lower the cistern and use a shorter flush pipe on it. The greater the distance between the cistern and the bowl the greater the chance of it splashing. As the water moves that little bit faster on the long flush pipes.
It could be one other factor though that the toilet bowl is not compatible with the cistern. An old cistern with a modern bowl is asking for trouble as these modern bowls are made to use less water to flush them so the old cistern just simply has too much water every,time it is flushed to actually escape down the drain so it backs up a bit and could be cause the water to splash. Cut the water level down as above or add a bottle to the cistern.
One thing I did forget is it the inlet valve spraying water onto the lid of the cistern and then hitting the seat or floor. If that is the case you need to put new rubber seals in the fresh water inlet valve in the cistern most hardware stores sell kits for these but be warned one kit does not do all, you have to find out what brand the cistern is and get the appropriate kit to fit that cistern.