Actually it's 100% the opposite.

Look at your iron-carbon phase diagram: https://www.tf.uni-kiel.de/matwis/amat/iss/kap_9/illustr/fe_...

Cast iron contains a high concentration of carbon, which makes it brittle. To make steel you need a heavy oxidizer to extract the (reduced) carbon. This is usually done using a "basic oxygen furnace", which is basically injecting a high pressure, very high velocity stream of oxygen into the metal to game thermodynamics into combusting the carbon without combusting the iron.