Take care that the base pose of your mesh is the same (A-Pose to A-Pose or T-Pose to T-Pose but not mixed). Afterwards: You could check if your twist bones make issues if you just skip them in the retargeting manager (don’t assign any twist bones of your mesh to the twist bones of the UE4 mannequin). Many animations works without them (like idle-walk-run and jump of the third person template). If you don’t have twist bones then you could get some twists that look like the twist of a balloon. For example if you hold a rifle with your left hand upwards. Without a twist bone this could easy look like you have too less flesh between your lower-arm and your hand (like a twisted balloon). It should look a bit twisted around the bone (too tiny radius) but should never look like a mutant-monster. Your animation looks like a mutant-monster so I’m pretty sure you have another issue.
It looks like your hands are almost at the right place (if this is some “aim with weapon anim”) but your upper and lower-arms does not follow your bones. So I would recommend to verify if those bones got the right translation retargeting. Means open your skeleton in persona, click on “Show Advanced Options” in the skeleton tree and verify if “translation retargeting” of all bones with weights (except pelvis and root) are set to “Skeleton” (check all those pull-downs that you get if you open the advanced options). If this does not help and you changed the rest pose of your mesh already to A-pose then doublecheck if you choosed the right bone in the retargeting manager for UpperArm_L and LowerArm_L. UpperArm_L should be the bone that is able to rotate the hole arm. If it rotates just a part of your arm like a twist bone would do then you most likely choosed the wrong bone. Same with LowerArm_L but without the part of the upperarm of course. If your twist-bones are some childs of your UpperArm_L and LowerArm_L bones (even if you have more than the UE4 one) and you skipped all twist bones in the retargeting manager then you should already get a much better looking result because if you choosed “Skeleton” all those unassigned bones should just follow it’s parent-bone. If you retarget to idle-walk-run then you should not see that monster anymore (probably some fingers would do some strange things if you differ too much from the rest pose of the UE4-mannequin).
If your mesh does not look like a monster anymore but you got some twists now if you aim with a weapon try to assign some twist bones again in the retargeting manager. If they don’t have weird angles your chance is good that you get some results.