The problem, Pennyfarthing, is that fast-growers don't know when to stop. I can't think of anything that grows fast to a moderate height and then stops. I think it just isn't possible.

How about a fence or trellis - or a combination of the two - to the desired height, with some plants placed in front of it? Then they can take their time growing in, and you can have perennials and climbers that aren't there all the time.

Beattie is right , there isn't a conifer or any other plant for that matter that will grow fairly fast to a moderate height then stop. ok, there are plants more suitable than Leylandii, but you will have to be prepared to do some selective pruning over the years to keep whatever you choose to the required and a manageable size.

I agree with the earlier comments. You will need to prune a tree if you want to control size.

A Thuja grow moderately. It will also tolerate pruning.

Pinus Mugo, if you buy a more mature specimen, will not grow to Leylandii proportions, then you will get size but not have to prune.

I like Picea Glauca - they are attractive, small, dense trees. A few of these would look nice in pots.

I would agree with all of the comments above. Trees, by their nature, grow at a steady, even rate until they reach their mature years. There are plenty of small trees and plenty of fast growers, but none that fit both categories.

In terms of a fast growing, only moderately tall evergreen for screening purposes, a bamboo would be your best bet.

But if you plant bamboo you'll need to bury (deep!) a root restricting membrane or you'll have an impossible to dig out, spreading thicket taking over the garden. Or grow the bamboo in a row of pots to make a screen.

^^
Actually, Beattie, that depends on which bamboo you plant. There are literally dozens of species that never run at the root, at least no more than a similar sized shrub would do.

Any of the very beautiful high mountain species - Thamnocalamus, Borinda, the smaller Fargesia for instance - would do the job nicely. Always buy and plant clumping, rather than running species.

So the answer is, don't plant just any old bamboo! I couldn't get rid of the huge area of it in my last garden. After a few decades it flowered and I thought "At last! It's going to die!" And it didn't - at least, not until several years later. I see now it's mostly dead, but new shoots are re-appearing in a few places. Not my problem any more. :-) I've just helped myself to lots of the dead canes.

Good advise regarding bamboos, either do your research, or go to a reputable specialist ...or both!!

Bamboos get such a bad reputation as garden plants, but it's really not justified...there are, after all, many hundreds of species, of all shapes and sizes. Unfortunately garden centres often only stock the ones that spread quickly, partly because they're the easiest to propagate.

It's a bit like condemning all trees on the basis of the behaviour of C. leylandii in the average back garden. As in all garden matters, it's about growing the right plant in the right situation!

Thank you all very much. What about common old laurel? I don't think I've seen a very tall laurel. But as a result of this advice I'm now wondering about a medium sized deciduous tree (after all, I don't sit out there much in winter!) I don't think rowans or hawthorns grow very tall do they? I like the idea of a couple of large pots of bamboo in front. I love the sound of the wind in bamboo but have never grown it because I know it can be invasive.
I shall look at picea glauca, Kildamorie --haven't a clue what they are yet!
Thanks again everyone--you've given me lots to think about.

Don't be fooled by common Laurel, I got some at the back of my place 20feet high although they are very old trees/shrubs will grow 2-3 feet per year easily once established and happy!

Thanks Bluespruce, 20 feet is a bit drastic but I might have to downsize in a few years so maybe I wouldn't need to worry about pruning and meanwhile I'd have an effective, if rather dull, screen. I would rather have something more bird-friendly though.

Hawthorn makes for a wonderful boundary type tree. Of course they do get large eventually, but, if you're prepared to keep them pruned when necessary then they are pretty much the best wildlife tree you can plant.

There's a good little article about Hawthorns below:

https://www.ecocharlie.co.uk/blog/2010/05/the-may-tree/

Thanks Ilex, I will read the article! I am keen to grow wildlife friendly trees as well as have my screen so a summer screen might be a compromise...

Other wildlife friendly trees and bushes are those found in traditional field boundary hedges. I realise that you really only want one tree, not a whole hedge, but there are other suggestions of wildlife friendly bushes and trees on the links from this page -
http://www.ashridgetrees.co.uk/SHOPPING-CATEGORIES/Native-Hedging

Thanks Beattie, I'll have a browse this evening. I might just stretch to 2 trees! Everyone has been most helpful and given me several ideas to think over.