In a medium saucepan, combine water, caster sugar, and fresh lemon juice. Heat over low flame, stirring constantly until the sugar completely dissolves, about 2–3 minutes. Bring to a gentle boil for 5 minutes then remove from heat and cool for about 30 minutes.

Cut ripe watermelon into cubes and place in a blender; puree until smooth, about 1 minute, ensuring no chunks remain.

Strain the watermelon puree through a fine sieve over a large bowl to remove pulp and seeds, using a spoon to press out as much juice as possible.

In a large mixing bowl, combine the cooled sugar syrup with the strained watermelon juice and whisk until well blended.

Cover the mixture and chill in the refrigerator for about 2 hours.

Pour the chilled mixture into an ice cream machine and churn according to the manufacturer's instructions, typically around 20–30 minutes.

Transfer the churned sorbet into an airtight container, smooth the top, cover, and freeze until solid, about several hours.