Nah, sunscreens are way more harmful than sun exposure itself. It's the sunscreen that will cause the cancer eventually.

Here's the chain of events: young patient gets bad sunburn, sees doctor. Doctor sympathizes and warns patient of dire consequences of not using sunblock 100% of the time. Patient agrees and mostly uses sunblock for several decades, forgets a few times and gets more sunburns. In fact, patient is way more likely to be exposed to powerful sunlight because of the invincibility conferred by sunblock.

Patient gets cancer from substances in sunblock, and feels terribly guilty for not using it because a few sunburns must have given her skin cancer.