Both remove() and insert() have a runtime efficiency of O(n) for both ArrayLists and LinkedLists. However, the reason behind the linear processing time comes from two very different reasons:
In an ArrayList, you get to the element in O(1), but actually removing or inserting something makes it O(n) because all the following elements need to be changed.
In a LinkedList, it takes O(n) to actually get to the desired element, because we have to start at the very beginning until we reach the desired index. Actually removing or inserting is constant, because we only have to change 1 reference for remove() and 2 references for insert().
Which of the two is faster for inserting and removing depends on where it happens. If we are closer to the beginning the LinkedList will be faster, because we have to go through relatively few elements. If we are closer to the end an ArrayList will be faster, because we get there in constant time and only have to change the few remaining elements that follow it. When done precisely in the middle the LinkedList will be faster because going through n elements is quicker than moving n values.
Bonus: While there is no way of making these two methods O(1) for an ArrayList, there actually is a way to do this in LinkedLists. Let's say we want to go through the entire List removing and inserting elements on our way. Usually, you would start from the very beginning for each element using the LinkedList, we could also "save" the current element we're working on with an Iterator. With the help of the Iterator, we get an O(1) efficiency for remove() and insert() when working in a LinkedList. Making it the only performance benefit I'm aware of where a LinkedList is always better than an ArrayList.