LeeG

Thin plastic will tear sooner than thick plastic. trash compactor bags are 2.5mil(1000ths of an inch). Cheap black garbage bags might be .9mil.
If you simply cram a large thin garbage bag into a pannier there's a lot of bundled up plastic that will collect water if the top of the bag isn't folded down neatly. That's the reason for recommending a kitchen sized bag.

If you have hard things in thin plastic bags and they are rubbing against hard surfaces it will wear through thin plastic. Use a bungie to compress and hold panniers firm to rack if anything is loose. Wrap hard things. Items that absolutely cannot get wet should be double wrapped and in a seperate bag. Simply putting a .9mil garbage bag in a pannier then tossing everything in it loose isn't good enough because eventually you'll start putting in damp things and pulling out dry things and one of those times the pile of plastic will have a few tablespoons of water trapped in it and it'll fold into the inside.

If you're anticipating non-stop rain have one pannier designated for retrieiving things while on the ride and leave the other untouched for dry goods not needed until you're absolutely out of the rain and dry.