This claim is exaggerated and misleading. There is not enough results of studies that prove that birth control causes cancer. There are many different types of cancer and there are multiple factors that can lead to the root of them. Stating that birth control causes cancer is extremely misleading and can guide people to fear their contraceptives.
Oral contraceptives do not directly cause cancer. However, some studies have shown consistent evidence that there is an increased chance of developing breast/cervical cancer in women who use oral contraceptives. According to Cancer.Gov, "An analysis of data from more than 150,000 women who participated in 54 epidemiologic studies showed that, overall, women who had never used oral contraceptives had a slight (7%) increase in the risk of breast cancer compared with women who had never used oral contraceptives." Nevertheless, there are multiple types of oral contraceptives, and data from the Nurses' Health Study found that participants who took the specific "triphasic" pill showed the increased risk of breast cancer. The dose of hormones in the this type of pill is changed in three stages over the course of a woman's monthly cycle (Cancer.Gov). Additionally, Cancer.Gov states that those who have been using oral contraceptives for over 5 years have a higher risk of cervical cancer, "One study found a 10% increased risk for less than 5 years of use, a 60% increased risk with 5-9 years of use. [...] However, the risk of cervical cancer has been found to decline over time after women stop using oral contraceptives." There is little evidence that oral contraceptives directly cause cancer.
There are main breast cancer risks that are not oral contraceptives. These can include, a personal history of breast conditions, family history of breast cancer, reproductive history, weight, age, and more. According to Mirakind.Org, "your risk for breast cancer is due to a combonation of factors - hormonal, environmental, & lifestylle factors."
Overall, there are many risks for cancer; however oral birth control is not a main risk for cancer. This statement is misleading and is exaggerated.