A turnpike was a Toll road. Sometimes the word is used for the booth where you paid your money to use the Turnpike.
No idea of any other meaning nor why there should be one in someone's garden.
There is a house with a garden for sale in a Turnpike Lane though.

Are you looking at a map and seeing the word written on what is now a garden? If so it is what Snoopdog has said. A turnpike in Scotland is an old spiral staircase which I doubt would be used in a garden unless you live in a castle.

Or its an exam question........

Mmmm... (Give reasons for your answer.) ;-)

A friend of ours had a turnstile in his front garden as a feature.......

A handy place to sell tickets for an open garden event!

Don't think ours would get many takers...