When we wake up in the morning, humans immediately enter into the world of time. We need to eat, get to work, keep clean, go places, do things, etc. We budget out the time of our day accordingly. But animals don’t do this. They eat when they are hungry or when food is available. They are not on a self-imposed schedule and do not get indigestion because they are caught in traffic.
Both animals and humans are conscious and aware of their environments. But only humans are aware of the passage of time. Therefore, human consciousness provides the basis of the sense of time. So in that sense we could say that human consciousness governs time. Time goes slowly when things are boring. It goes fast when things are fun for us.
But time can also govern consciousness in the sense that people are constantly aware of the fleeting nature of life. There is not enough time to get everything done that we want to do. There is a shortage of it to us. And therefore we have anxiety issues that run our lives. Time becomes the ruler of our thoughts as we oscillate between regrets about the past and worry about the future.