How to stop mapping on true results for large lists?

The AllTrue function will short circuit.

AllTrue[list, EvenQ] 

But I may have misunderstood, because you said you want short-circuiting on True, so maybe you want

AllTrue[list, Not@*EvenQ] 

which is baically equivalent to

AllTrue[list, OddQ] 

Anyway, you can craft whatever specific function you need.

You can use AnyTrue. AnyTrue short-circuits on first True result and returns True in that case.

With

list = {1, 3, 5, 2, 7, 9, 11, 13, 15, 17, 19}; 

then

AnyTrue[list, EvenQ] 

True 

Hope this helps.

alist = {1, 3, 5, 2, 7, 9, 11, 13, 15, 17, 19}; blist = {1, 3, 5, 7}; f = FirstPosition[#, _?EvenQ, False] /. {_} :> True &; f /@ {alist, blist} 

Result:

{True, False} 

Unlike the case with Map, Throw may be used to exit from Scan

Maybe

list = {1, 3, 5, 2, 7, 9, 11, 13, 15, 17, 19}; Catch[{Scan[If[EvenQ[#], Throw[True]] &, list], Throw[False]}] Catch[{Scan[If[EvenQ[#], Throw[True]] &, DeleteCases[list, 2]], Throw[False]}] (* True False *) 

list = {1, 3, 5, 2, 7, 9, 11, 13, 15, 17, 19}; TakeWhile[list, EvenQ] == list 

False

LengthWhile[list, EvenQ] == Length[list] 

False