Is my work as a website designer for a food company, some of whose products contain cartoon images, considered unlawful (haram)?

It is permissible to work in companies that produce products with disparaged images (such as children's sweets), because these images are not intended for their own sake and are discarded after opening the package.

Similarly, it is permissible to place digital images on computers or websites, provided they are free from شرعية (religiously forbidden) abominations, especially if there is a benefit in them for the producer or the consumer.

Regarding obsessive thoughts (waswas), it is advised to strive against them and not surrender to them, and to seek protection through remembrance (dhikr). It may also necessitate consulting a specialist.