I had to turf a similar sized area to you which had previously been a gravel area. I wheel-barrowed out around 5 tonnes of gravel. I then reinstated 6 tonnes of topsoil by barrow. I heeled then raked, then heeled again at 90 degrees then raked again (you get the picture!). I was lucky enough to find an old roller behind a shed would you believe, so I rigged that up and rolled it again. As I was adding quite a depth (6+"s in places) I had to make sure it was compact as possible to avoid subsequent problems. It was one of those jobs I wish I'd never started but a few months on it's looking great. I also had to dig out a load of woodrush hence the patches all over.
The end result looks amazing. The hard work clearly pays off. I very much doubt we'll be lucky enough to find a roller anywhere here but we can certainly give it a good go with a rake and stomping!
A good heeling-in and raking and repeat and repeat will do the trick. If you can borrow someone's lawnmower (with a rear roller) you can use that (with a bag of compost drapped over the engine for added weight) and push that around. Just an idea.
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https://www.gardenersworld.com/how-to/maintain-the-garden/how-to-prepare-the-ground-for-laying-turf/
Gardening in Central Norfolk on improved gritty moraine over chalk ... free-draining.