Is it normal to want to promote a new employee imedietly after hiring?

So I have worked for 5 straight years and often left in charge as I am considered the responsible one. However even in most of those instances have bwwn told by superiors I am not qualified to bw a leader or manager or run things. However with my past three jobs I came in and they wanted to promote me to management almost immediately. I am a little suprised becuase I did run crews and stuff but even though I been given the job of lead a lot of times am told that leadership just isnt for me. Is it normal to want to make someone a manager after a week?