Yes we do brew "beer" in that is what we call all drinks of that type, so bitter, lager and ale are all "beers".

The only American beers I have seen over here are Budweiser and Miller. They are "brewed under licence" so you can bet they bear little resemblance to what you get in the States Stuff brewed under licence tends to have been modified to suit what are perceived to be local tastes.
I have had imported Michelob (in bottles) at a US Navy base, and I found that rather nice

There was a trend over here in the 1980's and 90's for the large brewery chains to take over the smaller local brewers and foist their chemical crap on their outlets. Eventually this spawned a reaction in the formation of CAMRA (CAMpaign for Real Ale) who avidly promote traditional beers, brewers, and outlets.

Things have gone full circle now, with the emergence of small and micro breweries making "real ale" (organic) using traditional methods and ingredients. These are generally what I drink.

I know Sam Smiths, it's original brewery is about 9 miles from where I was born. It has been going since when the United States was still a part of the British Empire It has been independent and family owned throughout.