Raising height of a lawn?

I have a small lawn that gets very boggy in winter, it's on heavy clay with only a thin layer of topsoil. Sadly there is no way of adding drainage.
I'm considering raising the height of the lawn which will allow me to have a thicker layer of topsoil/sand above the clay. My question is should I..
Strip off the turf, lay it upside down, add topsoil and then new turf?
Strip off the turf and remove (stack elsewhere), add topsoil and re-turf?
Or, could I simply add topsoil on top of the existing lawn, then re-turf?
I'm considering raising the height of the lawn which will allow me to have a thicker layer of topsoil/sand above the clay. My question is should I..
Strip off the turf, lay it upside down, add topsoil and then new turf?
Strip off the turf and remove (stack elsewhere), add topsoil and re-turf?
Or, could I simply add topsoil on top of the existing lawn, then re-turf?
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If you remove turf and turn it, then add more soil etc, that's a fair old bit of extra height. Just adding soil wouldn't really work, unless you were adding a lot, so again, the height would be considerable.
I'd be inclined to remove the turf, dig over the remaining soil and add manure and pea gravel. Then some soil and turf. The former will help with drainage. You'd need to let it settle a bit before doing the turfing.
You could add stepping stones for border access but without drainage your garden will always be boggy.
Mine is too. In fact it's partially flooded from Nov through to early Feb..I've just learnt to live with it. I could add drainage but I don't want the mess that goes with it. It's the time of year I have a rest from gardening.
If the existing lawn is compacted it would be sensible to break it up a bit. If not you could probably get away with building up levels on top of existing. I'd probably mix sharp sand with topsoil to increase the levels (>50%). I don't think there's any benefit to adding manure or compost etc.
I work as a landscape architect on various housing developments, and poorly drained rear gardens on clay is a common issue; our solution has been to add 150mm of pure sharp sand, with 300mm of topsoil over that and then the lawn. The sand layer connects to land drains or soakaways at the back of the retaining walls running through the site. On your site that would be overkill, and you don't have a land drain to connect to, but I am thinking raising the lawn levels with an extremely sandy topsoil (rather than a pure sand layer) would probably help.