The yarn thickness / weight / TEX in combination with the size of tool (crochet hook, knitting needle, etc.) influence the size of the stitches.
If your yarn is thinner than the suggestion but you use the same size hook, it will be looser than the original. Since this particular shawl is a filet pattern, the actual dragon pattern might get lost in all the loose stitches. Choosing a bigger hook only increases the problem.
If you use a hook that is suggested by the yarn (which would be smaller in this case), your stitches get smaller as well. If you work according to the original pattern, the finished shawl will be smaller.
That's why there is a gauge given in the pattern:
Gauge: 8 stitches and 10 rows = 10 cm
That means to follow the original instruction you have to make 8 (horizontal) stitches be 10 cm wide and 10 (vertical) rows 10 cm long.
You can do that in 3 ways:
- Find thicker yarn. That's the most simple solution but not the one you prefer.
- Make your yarn thicker by using 2 strands at once. My experience with crocheting is limited, because I personally find it too fiddly, but doing it with 2 strands at the same time makes it even more fiddly. I would not recommend this for a beginner. If you're experienced and confident that you can manage, go for it.
- Use the thinner yarn but make more horizontal stitches and bigger vertical ones until you can reliably create a 10 x 10 cm swatch. If the original pattern asks for 4 stitches to create one filet square, try making 6, 7 or 8 instead for one square. If the patterns asks for double crochets to fill one filet square, try making a triple or quadruple crochet instead. Create a test swatch of 8 x 10 filet squares that is as close as possible to 10 x 10 cm and use the same number and type of stitches for the actual shawl.
The disadvantage of option 3 is that the stitches are still thinner in diameter, so the overall impression of the finished shawl will be more empty space than yarn.
The alternative hack: make smaller stitches but enlarge the pattern to fit your yarn.
Filet crochet is a pattern of regular squares. The filled squares become the motif (the dragon in this case). If your yarn is only half as thick as recommended, your squares will be about half as big. That means you need twice as many squares horizontally and vertically to create a shawl in the right size.
The easiest way to enlarge the dragon motif is to double it in size. If the pattern asks you to crochet one row with 1 empty, 3 filled and 2 empty squares, you crochet 2 rows with 2 empty, 6 filled and 4 empty squares instead. That changes the overall impression of the finished shawl to more yarn than empty space. It also makes the edges a little more blocky (like a badly pixelated image).
To smooth out the edges you need to copy the motif onto a sheet of squared paper, doubling it in size (4 squares on paper for each filet square of the pattern). After doing that, you smooth out the pattern by leaving single squares empty along the edges where filled and empty squares meet in the motif.
I hope this makes it clear: the light blue squares are the ones that should be filled if you simply double the size. Leaving them empty smooths out the contours.