Question: My toddler doesn't like to get his teeth brushed. How do I get him to be more cooperative?

He knows it's part of the bedtime routine and doesn't try to avoid it. He will happily grab and hold out his toothbrush for us to put toothpaste on it, and then he enjoys sucking and chewing it. 

When he's done that for a short while, we decide it's necessary to do it properly, and that often ends in me holding him while my wife brushing his teeth, while we explain what we're doing and why. It only takes 20 seconds, but it's always accompanied by screaming and squirming. I'd like to find a way to avoid this. As soon as we're done, everything is fine again.

**Clarification and update:**  

At this age, we don't expect him to be able to brush his teeth on his own. What I meant was that he didn't allow us to do it for him either. 

In the meantime, four months have passed since I asked this, and he's developed a lot. Now he gladly lets us brush his teeth carefully and cheerfully. 

**Update 2:**  

Another 6 months have passed, and he's still not enjoying the experience. He understands that it has to be done, and he sees us do it too. Sometimes he'll happily let us brush his teeth, other times it's a struggle. 

We are using several suggestions from the answers below -- we sing, we distract, we talk about "dirty teeth" and brushing the "dirty sugar" away, we let him brush on his own for a while and then take over, he can see himself in the mirror, etc.
All of this helps, but one factor certainly is that he's tired, and that's natural because brushing teeth is part of the bedtime routine.

Answer: 
Our daughter was the same with the sucking and chewing but one trick we use that has worked really well is one parent brushing at the same time while the other parent is the 'judge' and seeing who can make the loudest noise while smiling and brushing up and down/ side to side,

Once it became a game with a bit of competition she happily stands there brushing for quite a while and 'making the noise' ensures she does it properly.

..also sometimes she likes to watch herself brushing her teeth in the mirror - if you have a big enough mirror then you can stand beside him so he can compare how you do it against how he does it; it's our fallback when the noise game fails.
We said that otherwise "microbes eat your teeth" (3 years old)