Why is 他 put after 出 in 认出他来 in 女大夫很快认出他来?
认 (recognize/identify) is the verb and  出 is its result complement 

认 = to recognize

认出 = to recognize (and success)

We cannot insert the object between a verb and its result complement

Examples:


We cannot insert the object between '看见' and say '看(他)见' , we can only say '看见(他)'
we cannot insert the object between '听到' and say '听(声音)到' , we can only say '听到(声音)'
we cannot insert the object between '认出' and say '认(他)出' , we can only say '认出(他)'


I think the confusion come from the presence of '来' at the end 

If it was just '女大夫很快(认出他)', we can clearly see '女大夫很快(认他出)' is grammatically incorrect

But in '女大夫很快(认出他来)' '认' would be a single character verb and 出来(out) would be a preposition (away from inside)


[认][他][出来] = [verb] + [object] +[preposition (out)] which would be grammatically correct -- That's why you think '他' should be placed before '出', but the fact is, 他 could only be placed before '出来' , not '出' (see example #3 above) 
[认出他][来] = [verb + result complement+ object] + [final particle 来 (indicates an outcome)] which would also be grammatically correct - It is what we see in the sentence in question


More examples of  出来 as a preposition (out):

推(出来) = push (out)

拉(出来) = pull (out)

For the above two examples, you have to insert the object between the verb and 出来, as in "推(他)出来" = "push (him) out" ; "拉(他)出来" = "push (him) out"