Wash the persimmons, then remove the stem and peel them.
Quarter each persimmon and remove any seeds and blemishes if you see them. Then roughly chop the persimmons.
Place the persimmons and water in a large saucepan and cook it over medium heat, stirring frequently, for about 10 minutes.
Add the sugar, salt, and lemon juice to the saucepan, stirring until the sugar dissolves. Then increase the heat and cook until the mixture reaches a rolling boil.
Reduce the heat slightly, and cook for about 15 minutes, stirring frequently and skimming off any foam that forms if you wish. Remove the pot from the heat and if the fruit is still fairly intact, use a spoon to mash it up into more of a puree. Or, carefully puree the mixture using a stick blender inserted into the pot until it is very smooth (I use this method).
Now spoon a little of the fruit mixture onto the plate you placed in the freezer. Let it stand for 30 seconds, then run a spoon through the jam.. If it wrinkles, the jam is set. If it is still pretty liquidy, return the pot to the heat and cook for a few more mimutes, then retest.
Store the jam in clean, sterilized jar s in the fridge for up to 2 weeks, or follow the USDA's canning instructions (linked below) to keep your jam for up to 1 year.