Question: I have bought a second hand 19 inch frame and I am normally used to a 20.5 inch.

The bike rides OK but I would benefit from raising the handlebars.

I have read about fork steerer tube extenders and adjustable bicycle stems.

I personally think the fork extender would be a better way, are there any advantages or disadvantages to either?

Answer: 
A stem with a greater angle and rise, potentially coupled with a riser handlebar will be a stronger solution than a steerer extender. If you are doing casual riding on flat ground a steerer extender is probably OK. If you are riding rough terrain and putting a lot of force through the bars an extender may not be able to handle it.
Are you sure you have a threaded steerer? Like this?

[![enter image description here](https://i.stack.imgur.com/1wCRr.png)](https://i.stack.imgur.com/1wCRr.png)

If so, then there are a few options:

If you just want a quick bike fit fix and be done with it, then, as you most likely have a quill stem, get one with higher height and/or angle:

[![enter image description here](https://i.stack.imgur.com/EdXXn.png)](https://i.stack.imgur.com/EdXXn.png)

But, if you want a long-term thing and 'get with the times', you can fit a quill stem to threadless adapter:

(2nd image is an expander stem version, but it also works the same)

[![enter image description here](https://i.stack.imgur.com/lO4X0.png)](https://i.stack.imgur.com/lO4X0.png)
[![enter image description here](https://i.stack.imgur.com/VVqgb.png)](https://i.stack.imgur.com/VVqgb.png)

Though you will also need to buy an actual threadless stem too.