While there is *no* empirical evidence to indicate that transgender people pose any kind of threat to cisgender people in public bathrooms, the reverse is not true.
This was a large survey of 27,715
transgender people across the United States. The main bullet-points from their “Experiences in Restrooms” section (pp. 224–230) reads:
Nearly one-quarter (24%) of respondents said that someone had questioned or challenged their presence in a restroom in the past year.
Nearly one in ten (9%) respondents reported that someone denied them access to a restroom in the past year.
One in eight (12%) respondents were verbally harassed, physically attacked, or sexually assaulted when accessing or using a restroom in the past year.
More than half (59%) avoided using a public restroom in the past year because they were afraid of having problems.
Nearly one-third (32%) limited the amount they ate or drank to avoid using the restroom in the past year.
Eight percent (8%) reported having a urinary tract infection, kidney infection, or another kidney-related problem in the past year as a
result of avoiding restrooms.
“In our sample of transgender and nonbinary US adolescents, the 12-month prevalence of sexual assault was 25.9%, substantially higher than national rates of 15% among cisgender high school girls and 4% among cisgender boys. After adjusting for potential confounders, compared with nonrestricted youth of the same gender identity and sex assigned at birth, school restroom and locker room restrictions were associated with 1.26 times the risk of sexual assault for transgender boys, 1.42 times the risk for nonbinary youth AFAB and 2.49 times the risk for transgender girls.”