Mid April to May will be a better climate. The weather can range between 18 degrees to 5 degrees approximately . I am guessing you will get goo deals during March as Jan-March the tourists are less.
Istanbul and European Turkey experience hot summers and cold winters with snow being a common feature. Spring and autumn, from April to May and from September to mid November respectively, are the ideal times to visit Istanbul and the inland regions when temperatures are pleasant and the skies are clear.