'What's a betting shop? '
They're called 'bookies' here. Or 'the bookies'. Same as you have 'bookies' who operate mostly illegally in the US. Only there, it's usually lone operatives or lone operatives who are part of organised crime. Here, it's large corporations who have a chain of storefront shops where you can bet.
Gambling differs by state in the US, of course.
It is generally legal on Indian reservations, so in many Southwestern states, it's a source of revenue on reservations, as they open a casino which tourists can visit. Reservations can also sell cigarettes with less tax in some cases (this is true in my home state of Texas) so people travel there in hired buses/coaches for day trips to gamble and buy cigarettes.
Similarly, in some states, horse and greyhound racing, with betting, is legal.
Gambling is also legal as soon as you are in international waters, so in coastal towns and cities of states that border on the Gulf of Mexico you'll often find day cruises that offer gambling.
How do I know all this?
My mother is an ace poker player from a family of card sharks who earns money to travel about the globe without denting their pension through very calculated gambling. She also plays and deals on several card leagues, all legal so long as the wagers are not monetary or are for charity in our state, as she truly enjoys playing all manner of cards. She's an ace croupier, a real advantage when it comes to playing for money where it's legal.
She plans two, 4 day trips to Vegas, a city she and my dad first visited on their honeymoon in 1964, each year, with a budget of $2000. She never plays the strip. Always downtown or Henderson. She learned to appreciate single malt whisky, as drinks are comped in most places, but she won't play drunk. This way, she sips the booze and never goes over the edge, so she keeps sharp. She also smokes, which makes her spin out the drink even more. Her trick is to, with luck, win the game, then she calls the next one. She'll call an obscure one, her favs are High or Low Chicago, this throws most at the table off, particularly men (my dad will say, 'I don't like to play her because she calls all these feminine games), and increases her advantage.
Last year, after taxes, she came away with $26,000.