More than likely the water passages in your exhaust manifold are plugged. If the manifold hasn't been pulled and cleaned in a few years, it is most certainly plugged. You will have to pull it to unplug it.
When the engine is first started, and the engine is cold, much water is by-passed by the cooling passages of the engine and the exhaust manifold through the bypass line as noted by CrazyRu above. The thermostat should be closed during this time. This allows the engine to warm up faster.
After the engine is warmed up, the thermostat opens to allow cooling water to enter the cooling passages in the exhaust manifold, cooling the manifold, exhaust gases, and the engine. If the exhaust manifold is plugged, cooling flow is EXACTLY the same as if the thermostat didn't open.
You will need a thermostat housing gasket, a gasket for the exhaust manifold to engine connection, a spacer that goes between the manifold and the engine where the two aft-most studs for the manifold are, a gasket for the blanking plate on the forward end of the manifold, and a gasket for the flange (plate) on the aft end of the manifold where the mixing elbow attaches. If you have the 45 degree mixing elbow, and it is good shape, but it hasn't been off for a while, do not try to disconnect the elbow from the flange on the aft end of the manifold. That flange wasn't available as a spare last time I checked, and if you bugger it up, you may have to repair it or have a substitute made. Pull the manifold and mixing elbow off the engine, then disconnect the mixing elbow and its flange at the manifold connection, pull the blanking plate off the forward end of the manifold, and start digging out the corrosion with tools that won't damage your manifold. Replacements aren't available. Inspect it carefully before you start. They are prone to crack with age and the thermal stresses of their life, especially if they have been overheated and then cooled rapidly while still hot. Cracks can be welded or solder repaired.
The hardest part of this job the first time I did it was getting the mixing elbow and exhaust hose connection apart. I tried to take the hose off the elbow as the first step. Next time I did it, I pulled the manifold and exhaust elbow off the engine so I could pull it forward off the hose. MUCH easier.
Hope this helps.