Nature is rarely binary. Sex and gender are no exceptions to that. The links provided below are to peer reviewed research, statements from medical organisations, historic data and other expert information sources. They will step you through the established facts around sex and gender.

Intersex people exist. There are plenty of sex variations to prove that sex is absolutely not binary, although being assigned male or female is typical. Read the following for more information on sex:

• Classifying Sexes - https://www.digest.ugent.be/article/id/84674/ 
• 10 Questions and Answers about Intersex - https://thisisintersex.org/advanced/10-answers-to-questions-about-intersex 
• What is Intersex - https://isna.org/faq/what_is_intersex/
• Beyond X & Y: Chromosomes and Sex Organs - https://scienceline.org/2020/10/beyond-x-y-chromosomes-and-sex-organs/

Gender exists. Most peoples gender matches that which is assigned to them at birth (cis gender). However there are a smaller number of people whose gender does not match that which is assigned to them at birth (trans gender). Trans gendered people have always existed. Read the following for more information on gender:

• What are the other 72 genders - https://www.medicinenet.com/what_are_the_72_other_genders/article.htm
• Trans and Gender non-conforming histories - https://historicengland.org.uk/research/inclusive-heritage/lgbtq-heritage-project/trans-and-gender-nonconforming-histories/
• 16 Remarkable Historical Figures who were Transgender - https://historycollection.com/16-remarkable-historical-figures-who-were-transgender/

There are currently over 3000 research papers published worldwide providing evidence that trans women are women, and trans men are men. Here are just a few:

• Evidence Supporting the Biologic Nature of Gender Identity - https://www.sciencedirect.com/science/article/abs/pii/S1530891X20357682
• Endocrine Society Position Statement on Transgender Health – https://www.endocrine.org/advocacy/position-statements/transgender-health
• The Neuroanatomy of Transgender Identity: Mega-Analytic Findings From the ENIGMA Transgender Persons Working Group - https://www.sciencedirect.com/science/article/abs/pii/S1743609521004252

The bottom line is that trans women are women. To say any different is offensive and biologically incorrect. Once you have that as your base, then which restroom to use becomes simple. A trans women uses the women’s restroom and trans man use the men’s.

Trans women are not a risk to cis women in single sex spaces. However, trans women are at risk from cis people. There is plenty of established evidence in this area, which contradicts the false narrative from gender critical organizations. Here are a few:

• 15 Experts Debunk Right-Wing Transgender Bathroom Myth - https://www.mediamatters.org/sexual-harassment-sexual-assault/15-experts-debunk-right-wing-transgender-bathroom-myth 
• The Transgender Bathroom Debate at the Intersection of Politics, Law, Ethics, and Science - https://www.researchgate.net/publication/326547236_The_Transgender_Bathroom_Debate_at_the_Intersection_of_Politics_Law_Ethics_and_Science
• Gender Identity Nondiscrimination Laws in Public Accommodations: a Review of Evidence Regarding Safety and Privacy in Public Restrooms, Locker Rooms, and Changing Rooms - https://escholarship.org/uc/item/4rs4n6h0 
• School Restroom and Locker Room Restrictions and Sexual Assault Risk Among Transgender Youth - https://publications.aap.org/pediatrics/article-abstract/143/6/e20182902/76816/School-Restroom-and-Locker-Room-Restrictions-and?redirectedFrom=fulltext

In 2015, there was a large scale survey of transgender people in the US (27715 trans people - https://transequality.org/sites/default/files/docs/usts/USTS-Full-Report-Dec17.pdf). 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.

BTW. Trans is short for transgender. It is a descriptive word. E.g. tall woman or short woman. When you talk about cis, it is also descriptive. All trans women, cis women, tall women, tall overweight women are women.