Usually the tenseness starts in the shoulders. Ride thinking to relax the shoulders, then the elbows and then hands, then do it again and again and again until your body understands how it feels to relax. Takes time and consistency for you to change habits.
Sorry I meant just the outside hand and rein holding on. Not both in one hand!
I do it too (I think most people struggle with it at one point in their career). I was told to hold my saddle pad and rein in one hand. People also make "monkey bars" on the saddle by clipping a leather strap to the saddle where a breast collar normally attaches to hold on to. I found it to help a lot and it will make you apply even pressure which helps your horse get a better feel in the bit.
In terms of what amber said, a driving rein will allow you to better see just how much your hands are moving, and won't let you use your ring ringer for leverage.
You probably need to relax your whole body and arms, and seat. Try to feel yourself sinking into your seat, and take deep breaths. With every breath, imagine your muscles relaxing, and your whole body becoming more elastic. This will help your elbows better follow the bit. You can also over exaggerate the following, back and forth movement, until you get a better feel for it.
I have the same problem, but have heard that using a driving rein helps