Why is Mrs Putnam so quick to believe that Goody osburn is a witch?

Answer 1

Because Osburn was the midwife when three of Putnam's children died. Putnam is so crazed with grief, and she blames herself for her childrens death, so she leaps upon any excuse to believe that an outside force took away her children.

end of Act One:

Mrs Putnam: I knew it! Goody Osburn were midwife to me three times. I begged you, Thomas, did I not? I begged him not to call Osburn because I feared her. My babies always shriveled in her hands!