Does this exist (camping related)?

[KenAdams]

And if so where/what should I search for?

I have a tent and am looking for somewhere to camp for the night. I don't need any facilities so it would be like wild camping, but I want to be the only tent there rather than a heavily populated camp site.

Private camp sites bring up ones that sleep 30 for a party etc or they are private but already have a yurt or something on them.

I'd just like somewhere pretty where I can set up my tent for the night then pack up and leave the next morning. The tent is quite large so we can't do actual wild camping as they usually only allow smaller tents or bivvy camping.

There are no campsites that have only one pitch that I know of in England. You might be able to wild camp in Scotland in a national wilderness as it is legal there last I looked.
But in Wales/England, A farmer may let you pitch on his/her land. But you’d need to ask permission.

In mainland Europe there are many many wild camping places where you can do what you want though.

There are some on air bnb

Choose where you want to go.
Find the contact details of a local farm.
Ask them.

Is this for a one-off, or a regular thing? If you join the Camping and Caravanning Club, they have lots of sites without many facilities.
www.campingandcaravanningclub.co.uk/membership/

The camping and caravaning club have a group of sites which they call 'Certified Sites' . They're all small - 5 pitches or less and usually very basic - Tap and a toilet if you're lucky.
We use them more than the popular well serviced sites and more times than not we're usually the only ones on site.