Ideally, the ice and water should reach an equilibrium at zero celsius. But this equilibrium might take a long time to happen, based on the exact setup.
Looking at a typical setup of yours, the ice will float at the top of the container and there is water (but no ice) at the bottom. Water is densest at $\sim 4$ degrees celsius, and such water will sink to the bottom of the container. The water at the top is in contact with the ice and so, it must be near zero celsius, or getting cooler.
So I imagine a situation where we have ice and cold water on top, near the surface (which has to be in equilibrium with the atmosphere, but let us neglect that for now). There is also cold water at the bottom of the vessel (at 4 degrees celsius or cooler), with a temperature gradient leading to the top. The actual temperature profile will depend on all kinds of non-equilibrium physics having to do with the shape/size/conductivity of the container, the size of the ice cubes, etc.
My guess is that since you would have submerged the thermometer a significant depth into the container (maybe close to the bottom), you're measuring the temperature to be closer to 4 degrees celsius than to zero celsius.
If my hypothesis is correct, then by stirring the contents of the container, you should be able to set up convection currents which will cool the contents more uniformly. You might want to be careful in not stirring the container too vigorously, for that will heat up the contents. I think gentle stirring should do the job without infusing much heat into the system, at least at the level of accuracy of your measurements.