When you put water in a glass full of ice, the water does not turn into ice because the temperature of the water is higher than the freezing point of water. The freezing point of water is 0 degrees Celsius (32 degrees Fahrenheit).
When the water is at a temperature above the freezing point, the ice in the glass has a lower temperature than the water. Heat flows from the warmer object (the water) to the colder object (the ice) in an attempt to equalize the temperatures. As a result, the ice absorbs heat from the water, causing the ice to melt.
The transfer of heat continues until the temperatures of the water and ice reach an equilibrium point. At this equilibrium point, the temperature of the water and the remaining ice stabilizes, and any further melting or freezing stops.
So, in short, the water does not turn into ice because the heat from the water causes the ice to melt until an equilibrium temperature is reached between the water and the remaining ice.