+1. Semi's are lighter weight, too. I have a Winchester 1300 20 gauge. Both my father's Remington 1100 12 gauge and my polychoked Saiga 12 are lighter than it.
You should really put some guns on a scale instead of just guessing. I don't know what a 1300 in 20 ga weighs, but a 12 is somewhere between 6.5-7 lbs depending on the exact version, barrel length, etc. and is the lightest pump ever made that I'm aware of.
You need to look at some manufacturers specs., but as a a rule pumps tend to be lighter. If you can find a Winchester 1300 it will be the lightest, with a Mossberg 500 just a bit more. Around 6.5-7.25 lbs depending on barrel length, stock etc. A Remington 870 will be around 7.25-7.5, as will the Mossberg 835. The Browning BPS, and Benelli Nova's are pretty heavy at around 7.75-8.25 lbs or so.
The Benelli' autos are the exception and will be very close to 7 lbs or so, but most gas autos are at least 7.5 lbs up to 8.5.
A lighter gun will kick a bit more, but I prefer to save the weight. My prefered gun is one of the 7 lb Benelli's, but I often carry an 870 as well at times. A lot depends on were, and how you hunt. I walk a series of old logging roads on public land and often end up walking 10-12 miles in a day of turkey hunting. I figure I'll shoot only once or twice a season anyway so I'll live with the recoil. The new recoil pads really equalize things so even the hardest kickers are a lot easier to deal with. The last bird I killed was 5 miles from my truck so a lighter gun along with a 15 lb bird made for a lot easier 5 mile walk. I'd already walked 6-7 miles before I shot the bird. If you have a short walk, then weight is not a consideration other than recoil management.
There is really no right or wrong answer. Pumps tend to be cheaper and are usually lighter, with a few exceptions. Multiple rapid shots are pretty rare, so there is nothing wrong with using a pump as a dedicated turkey gun. As an all around gun I prefer semi's, but use both depeding on the situation.