I bought an area rug to put in my living room, on our carpet , but it keeps turning up and moving
They do have double stick tape for rugs Judy. You might try it.
Here's a link with a few ideas, Judy:
Turn upside down and put weights on it for a few days.
I place a thin anti slip rug pad between the two. You can pick them up for very little money on Amazon or discount stores.
I use this type of double-sided carpet tape on all my clients carpets, it is especially made for sticking carpet on carpet.....its usually sold in the home store carpet area.... https://www.amazon.com/NEW-Original-Technology-Adhesive-Hardwood/dp/B01LYMTCHG/ref=sr_1_11?crid=1WTPK7SX01IY3&keywords=carpet+to+carpet+tape&qid=1576297266&sprefix=carpet+to+carpet%2Cfashion%2C266&sr=8-11 I first stick to the bottom of the area carpet/or runner, then remove the plastic and stick it down and stomp on it a bit to get it to adhear.
A non-slip rubber shelf liner for the dollar stores work very well, they come in a good long piece. Just sandwich it between the 2 rugs.
On the back of the rug, use a hot glue gun and make lines of glue about an inch from the edge all the way around. Let it dry. Flip the run over and the glue strips will act as a gripper for the rug.
For Large rugs, you may need to make several rounds of glue.
You can buy a rug pad from amazon or target!
These work very well! Just pick your size!