You are correct, it is a lay-by in British English.

In the US, 'turnouts' are generally restricted to mountainous areas--i.e. areas where a slower car/semi would be able to pull into to allow faster cars to pass. You could also generally refer to it as a 'shoulder.' The 'shoulder' is a part of the road that is off to the side and is used for cars to pull over safely. With 'shoulders,' generally it is not a legal lane, so it is slightly different from turnout lanes.

As an American, it’s not something often seen and I’ve never heard the term ‘lay-by’. ‘Turn out’ seems like the best general term but you also might hear ‘pull out’. It also looks like a ‘stopping area’, or ‘parking area’, depending on the rules for it.