I always need this when I buy stuff from the market. So, I need to know the real name for it. I always call it "bag" but it seems that it is not the correct name.

I always need this when I buy stuff from the market. So, I need to know the real name for it. I always call it "bag" but it seems that it is not the correct name.

In my experience, which is all I can answer from, in American English, 99% of the time the white plastic object is called a bag.
Specifically, it is a plastic bag.
We also have paper bags. And canvas bags.
When you start talking about "bags" made out of thicker material, traditionally woven from cotton or something, you can then use the word sack.
You can use sack to refer to a paper bag, but it is rare to refer to a plastic bag as a plastic sack, in the USA, using American English, at least in the dialect I speak.
So, in short, yes, bag is the most common term to call the white plastic object.
I have a funny experience with this. I grew up in the Eastern U.S. What you ask about were called bags, pure and simple.
Then I moved about 1500 miles (2400 km) away, to the American midwest, near the geographic center of U.S. I still remember my puzzled look when the cashier asked me, "Would you like a sack with that?"
Where I had grown up, a sack was made of burlap or mesh; i.e., this was a sack:

Fig. 1: A sack in the Northeast
But in my new location, your groceries were put in a sack, unless you wanted to carry them back to your car in your arms.

Fig. 2: A sack in the Midwest
By the time I moved back to the east coast more than a decade later, my ear had become accustomed to sack, and bag seemed eerily unfamiliar. I was just about certain that bag was the right word to use again, but it didn't sound right to me. So, maybe a month or so after I had moved back, I had this rather amusing conversation with a convenience store clerk while purchasing a gallon of milk:
J.R.: If I wanted one of those brown paper things to put my milk in, what would I ask for?
Cashier: A bag?
J.R.: Thank you.
Cashier: [looking confused]: Do you want one?
J.R.: No – I just wanted to know what it was called.
and then I walked out.
I think the bottom line is that regional variations may apply. I'd say bag is usually right, but you can always ask a cashier if you're not sure :^)
By the way, one question often asked in groceries stores is, "Paper or plastic?" I'd never really thought about it before, but that terse three-word question avoids the noun – which might be by design.
Lastly, remember this: If you are ever asked "Paper or plastic?" You'll have to decide for yourself. Why? Because baggers can't be choosers. ;^)
This appears to be UK-English only
It's a bag; specifically it's a carrier-bag.
You could ask for either - or even, colloquially, just 'a carrier'.
Some places would ask whether you want paper or plastic, though that's not a popular choice in the UK.
Here they prefer you to have a thicker, heavier plastic or cloth bag which can be used over & over, often referred to as a 'bag for life'
Examples -

In Br Eng, this would never be a sack - that would imply it being much larger, perhaps like this, a 'bin-bag' or sack.

In India, we refer that as a 'polythene bag'. Some years back, it was just a 'bag' but then due to the awareness about preserving our planet, the governments started acting strictly. All the newspapers and TV channels were flooded with the message that we should not use 'polythene bag' (there is when general public came to know the 'full form' of that bag!) However, polythene bags above 40 microns are allowed.
There are so many types of polythene but broadly, we refer these bags as polythene bags. The vendors here started using 'paper bags' and there is where, we, as buyers, have to specify the 'bag'. Suppose if I'm carrying something wet, I'll have to ask for a polythene bag over a thin paper bag that the vendor is using for everything else.
I would not prefer to put an adjective to this (such as shopping bag, grocery bag) which would simply restrict its usage! Use a polythene bag to carry a wet shoe, a fruit, a bunch of newspapers or anything that you don't 'shop' as well!
[I'm tagging this answer as InE].
In India wee call it a 'plastic-cover'. Well educated people also call it a 'carry-bag'.These are available in almost all types of stores in my country. In-fact because of its heavy usage and pollution problem a law was created in my state that demands these bags be made of plastic of 30 microns or above. And because in our country every state has its own language this is mostly known by it local name, in my state Kerala it is known in Malayalam as 'sanjhi'
I am a product specialist in the flexible packaging industry.
The bag you have in this picture is commonly called a "t-shirt bag". It is called this because the construction moderately resembles a t-shirt. This type of bag falls under the larger category of "carry-out bag" which is any bag that a cashier will put your item in at the time of purchase and you will then carry-out of the store. The application is important when it comes to figuring out if the item is subject to import tariffs..
They are made out of LDPE or HDPE. The bag in this picture also has registered spot printing done with a flexographic press.