It is more the US that had big dollar counterfeiting issues, to the point where a lot of places would only accept cards.
I think counterfeiting is probably low on the list of reasons why many places only accept cards here in the US.
In no particular order:
A: credit card issuers are all competing with each other to get customers to use cards instead of cash. They do this by offering "rewards," which used to be points to be used for purchases through "partners," but now is increasingly just either a 5% discount at point of purchase (Target does this with their store-brand "Red Card), or some percentage of "cash back," commonly 1%, but for certain "categories" like groceries and gas station purchases the cash back can be 5%. Of course the categories change frequently, so you need to be on your toes to take maximum advantage, but cash back is always better than "points" you have to redeem for products.
YES -- it is obvious that cash back isn't free. It is paid out of the swipe fees charged to merchants so they can accept card payments. Of course the merchants have to charge 3% to 5% more for their products just to cover the cost of the swipe fees, and sadly the merchant agreements between the card issuers and the retailers generally precludes the retailer from offering a cash discount. So from a consumer perspective, you're paying for the swipe fee, anyway, so you might as well use the card and get the "benefits."
So to that end -- I will pay for purchases at my locally-owned coffee shop or record store or guitar shop in cash, and they get the benefit of not having to pay the swipe fee. For the big box chains? Screw 'em, I'm going to use the cards that give me the benefits.
B: some merchants see the cost of swipe fees as being lower than the labor cost to manage counting and handling cash. I asked my friend who runs a local coffee roaster/shop about this, and he said from his small perspective, he goes to the bank regularly anyway so dealing with cash is no big deal. And he appreciates the savings from not paying the swipe fees. But, with the big box stores, it's a lot easier for all transactions to be electronic so they don't have to have the armored car come and take a million dollars in cash to the bank.
C: Employers do not trust employees to not steal cash from the till, even though the tills are counted out at the start of a shift and counted in at the end. (Employees don't trust employers, either; different issue.) Employers will embrace any idea that keeps workers from handling large amounts of cash.