Tug Boats are the Beast. They pack huge power in small size.
Normally they have towing power of minimum 40T to 120T.
The credit actually goes to water. Water hardly provides any opposite force for Tugs to push or pull big ships.
Even a small 20T Tug can easily tow a standard panamax. only thing is that it will happen at very slow pace. The Tugs have one job, they have to churn enough water to create thrust which can displace or move ship. Remember, anchored ships are easily moved by sea waves, Tugs are way more powerful than that.
When a ship is anchored, normally it moves in circle because of the waves, instead of staying put at one place.
All the Tug has to do is break the inertia of the vessel and start the movement. Once this starts, it's an easy job from there.

Getting a large vessel moving is a tough chore but once in motion, stopping them is impossible in a short distance. Even if they are under their own power and put the screws in full astern, they will travel a very long distance before coming to a stop.

I may be talking out of my posterior here but I always thought that, rather than actually pulling or towing large ships, tugs were used more to guide the vessels - pushing from the side(s) to make sure they go in the correct direction in limited spaces.

I assume that this would require much less power than propelling the ships, which would be moving largely 'under their own steam' anyway.

I have always thought it was a bit of both but then I might be talking out of my posterior too.

Actually, a lot of interesting things come out of my posterior.