Net immigration was over 300,000 last year. You may need to build 100,000 new homes each year for a long time to keep up with demand. Or you could control immigration.
I don't disagree with your idea that immigration should be controlled, but feel it is slightly unfair to blame the shortage of housing on this. This problem has been manifesting itself for at least 3 decades (and duly ignored by successive governments whilst prices and therefore stamp duty rocketed).