Did the Nazi's start to kill Jews before World War 2 started?

Yes.

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The above answer is the popular view - rejected by most scholars since the 1960s. There is considerable evidence against it, however. Until August 1941 Jews were actively encouraged to leave Germany and German-occupied territory for wherever they could get to. They were even able to pay their fares to, say the U.S. or Uruguay, in advance in Germany ... This does not 'sit easily' with the existence of a master plan from the outset to exterminate them. In August 1941 there was a change of policy and in September 1941 all Jews under Nazi control had to wear the Star of David when outside the home, and the first regular deportations from Germany began the following month.

Please see the related question.