It's actually hane (from the Persian - I think it means basically "house" there), but -ahane gets reduced to -ane, so you have hastane (hospital), postane (post office) from hasta, posta but hapishane "prison" from hapis.
I think you even find the "full" forms such as postahane in older Turkish.
An agglutinative language is one in which words can be extended by adding lots of endings.
English usually only does this one or two deep (e.g. boat, plural boats; individual+ist+ic+al+ly = individualistically is already stretching it a bit).
Turkish is very free with adding endings after ending, e.g. the famous Çekoslovakyalılaştıramadığımızdanmışsınız "you were (apparently) one of those whom we were not able to Czechoslovakianise" from Çekoslovakya + lı + la + ş + tır + a + ma + dık + ımız + dan + mış + sınız.