The news can be particularly bad about contextualizing stuff like this. The World Health Organization's IARC is expected to list aspartame as "possibly carcinogenic". Here's what that means. It is one of the groups with the weakest evidence. It basically means there is some animal or mechanistic evidence of a carcinogenic effect.
But most importantly IARC doesn't take into account risk, just hazard. It doesn't account for dosage or exposure. So this grade can occur in dosages or routes of exposure that are unrealistic in daily life. There's another WHO committee that does actual risk assessments called JECFA. This committee previously has said that aspartame is safe in the dosages that people are likely to encounter in daily life (up to 40 milligrams per kg of body weight - that's about 15 cans of diet coke per day for a 150 pound person). JECFA is expected to release a new report on aspartame mid-July.