splitting lists into sublists

You can use ReplaceAll with an appropriate pattern in such cases

{{a, X, b}, {c, d, e, X, f}, {h, j, X}} /. {begin___, X, end___} :> {begin}

Note the use of named patterns of the BlankNullSequence variety that can stand for any sequence of zero or more expressions

lis = {{a, X, b}, {c, d, e, X, f}, {h, j, X}};
TakeWhile[#, # =!= X &] & /@ lis

{{a},{c,d,e},{h,j}}

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From comment by Mr. J.M

First[Split[#, #2 =!= X &]] & /@ lis

Using Position

lis = {{a, X, b}, {c, d, e, X, f}, {h, j, X}};

#[[1 ;; Position[#, X][[1, 1]] - 1]] & /@ lis

(*  {{a}, {c, d, e}, {h, j}}  *)