Yes, yes and I also (you can laugh) carry a bucket and a bottle of warm water and some soap in the boot. When I get home I make up a bleach solution and dip everything before it comes in my house, including the bags.
Here's what I do.
Pack bags, bucket, dettol spray, warm water and soap into boot.
Drive to supermarket
Leave phone in car. Phone is a vector.
Whilst in car, mask and glove up. Car keys in pocket, contactless card in pocket
Enter supermarket with bags, shop and bag up.
(here is where it gets tricky)
Then, I load the bags of shopping into the car, return the trolley, and CAREFULLY bin the gloves.
Walk back to the car. Dettol spray boot handle and shopping bag outsides, contactless card and car keys..
Pour warm water into bucket, thoroughly soap hands and arms.
Remove mask, dettol spray outside of mask.
Soap hands again, and face.
Drive home, walk into house, wash hands, prepare bucket of bleach solution (20ml per 5 l), take it out to front garden along with other, already clean bags. Wear home gloves.
Unpack bags, dipping every item in bleach solution, and without rinsing pack into clean bags. Porous items (cereal boxes) get wiped with bleachy cloth.
All used bags get dipped and packed together.
Go inside, wash hands and face (bleach splashes).
Then back outside, bring everything in, pat dry but don't rinse, put away.
This may seem extreme. I don't care.
Gloves and masks don't work WHEN PEOPLE DON'T USE THEM PROPERLY. Otherwise they work better than not having them.