The weather in Istanbul isn't that much different to London. Average daily maximums around 13C (London = 11C), falling back to about 5C overnight (London = 2C). Monthly rainfall for March = 61mm (London = 61mm).
http://uk.weather.com/weather/climatology/TUXX 0014.htm?dayofyear=80
The southern coast of Turkey is warmer but still not 'summery'. Antalya has typical daily maximums around 19C, falling back to around 9C overnight. However, it's wetter with around 102mm of rain in March.
http://uk.weather.com/weather/climatology/TUXX 0004?dayofyear=80
The only destination where you're likely to get reasonable prices and family friendly facilities in March is the Canary Islands. (23C by day, 16C overnight, with monthly rainfall of just 10mm in March)
http://uk.weather.com/weather/climatology/SPXX 0210?dayofyear=80
The most popular (but more expensive) alternative is probably Egypt:
http://www.tripadvisor.com/Travel-g297555-s208 /Sharm-El-Sheikh:Egypt:Weather.And.When.To.Go. html
Chris