This is called rippling and it's how the timeline was designed to work.
In most cases, when you trim off the beginning of your video, you want to create a new starting point. And, when you insert a clip, you want the other clips to move aside to let it in.
As I say in my books, you can override both of these effects by holding down the Ctrl as you add new clips or trim an existing clip.
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For this, the easiest way is to hold down the Ctrl key, as you drag the Asset to the Timeline, to Insert Edit, and not Ripple Edit.
[Edit] I now see that Steve beat me - had to answer the phone!
Thanks to both Steve and Bill, I feel like you are raceing to answer my questions.
Yes, but Bill is on Mountain Time, so I have the advantage!
Yes, as I was supposed to be on vacation, I even got up early to beat Steve, but he had some investment banker call, at just the time I was going to post!!!
Actually, even late at night, he normally beats me to the punch.