Yes, it sounds perfectly normal to me. Being appointed a leader of something may seem like a happy thing, but with all the joy it brings with it, comes a certain level of ownership and responsibility.
From the limited context, I am assuming that the speaker might have been taking his work lightly, knowing that any slack that he would've cut, till date, wouldn't reflect poorly on him. Now, that he has been given a leadership role, he must have realised that all his actions would be accounted for, and that he is responsible for whatever his group or section does.
There is a practice in many industries, where a person who generally takes his work too lightly, would be given more responsibility, probably for a short duration, which may create a positive change, or once and for all, prove his incompetence.
The vested responsibility on his shoulder must seem like a test of fate, which could burn or boost his work profile, which must put him in a state of anxiousness and pressure.