Why does the Car tire shake?

I'll bet it's not the tires, rather it's what connects the tires to the car. You may have tie rod, ball socket abnormality or control arm problem. If the tire is moving wildly in this turn condition, tire may be about ready to come off, so I'd be careful not to do the extreme turn and get it checked out.

Could have been caused by hitting something, age/wear, etc. there are covered joints that can open up and get rocks sand in there and quicken the wear.

We need to know the year, make and model of the car. Is it a rear or front wheel drive car?

The answers would come flying in if you had entered this information.

Yes

usually when tires shake its because one of the tire has low air pressure. get a tire pressure gauge from autozone for 99 cents and make sure the air in your tire matches the same air pressure.

worn front end component, or tires rubbing in the wheel well