The history of the potato is rather complex and full inconsistencies - perhaps not unexpected considering the period when Westerners discovered this now important vegetable.... during their conquests and explorations of the Americas from whence potatoes originate.
Accounts of Columbus' first voyage to the Americas mention that the Indians served up a cultivated boiled root 'not unlike chestnut in taste'. By the late 1500s potatoes were being cultivated in Spain and Italy, reaching Vienna in about 1588 via Italy. In the same year Clusius - in Holland - mentioned that he received a couple of tubers but was uncertain whether they had originated in America or Spain, and Gerard appeared to be growing something similar . In fact you get the feeling that every botanist and his wife was experimenting with potatoes. However, it is uncertain whether these 'potatoes' were the same as that of the Solanum type, or were what was known as Papas perunanum which had its origins in Peru.
Sir Francis Drake encountered potatoes of South American origin during his world tour in 1578, and it is possible that during his piratical expeditions to the West Indies around 1585 he may have acquired 'papas' samples and handed these to his old friend Sir Walter Raleigh who had farms in Ireland, near Cork. On the other hand there is another school of thought which suggests that possibly Sir John Hawkins brought potatoes to England in 1565. It is also suggested that Raleigh himself brought potatoes to Ireland after returning from Virginia - unlikely since he never ventured there. A much more probable candidate was Thomas Heriot working for Raleigh and who was sent to the Americas to see what the land had to offer. Mitchell  thought that Heriot had given samples to Gerard, and to Clusius who introduced them to Italy and planted them in France.
There are references to 'Batatas' being found in India [Van Linschoten - 1583] being: "somewhat red in colour... but sweeter of taste," however it is thought that the potato wasn't known about in the Low Countries until around 1620. One reference [Houghton - 1699] says that there was a longer and more succulent potato brought into England from Spain. Potatoes did not initially catch on in England and Miller  mentions that the vegetable was despised by the rich, but then we find Sir Hans Sloane - the botanist and naturalist whose personal collection of curios became the starting point of the British Museum - saying in 1707 that "Many live on the Irish Patatas, a sort of Solanum..." [the reference to Ireland reflects the work done on the early cultivation of the plant on that fair isle.]
Spinach is thought to have originated in the Middle East, being introduced to Europe in the 16th century. The much tougher New Zealand Spinach [Tetragonia expansa] was introduced from the Antipodes as an alternative crop in times of drought.
Another plant originating in the Americas [Central]. Brought to Europe by the Spaniards who made great use of it as a food plant, but unknown as to when it finally arrived in Britain. One writer [Acosta - 1588] describes how 'tomates' were 'good to eate', full of juice, and gave good taste to sauce.
Turnip is another of those vegetables with a very long history of cultivation and seems to be of temperate Western Asia and European origin, with many varieties emerging over the centuries.
If you are interested in edible wild plants and their possible uses [many were used in old times as crops], take a look at the WILD FOOD SCHOOL microsite [opens in a new window].