I just got a level 2 turnip and I don't know how I got it. I would also like to have a level 3 turnip next. From the request, I heard you use a sickle but I didn't quite catch what Arthur said as I speeded the text. Thanks ahead :)
This relies on other factors as well, though. Using your sickle on a crop doesn't instantly level it up. Placing the crop in a healthy plot is important. Most of the time, your soil will require above average levels of health, growth bonus and growth modifier. It sounds complicated, but you can see all of this stuff with the magnifying glass. It's incredibly useful. Buy it, love it.
At higher crop levels, you might get stonewalled and it will stop leveling up... This means your land needs improving. Use the glass to check which aspect needs attention. Using things like greenifier, 4-leaf clovers, formula or withered grass can improve these things. If you take care of the soil, it'll yield high level crops gradually, so long as you keep harvesting it with the sickle.
Playing: League of Legends, Rune Factory 4, Tales of the Abyss 3D, Tales of Xillia