I'm sorry you're feeling anxious about talking to your family after coming out as bisexual. It's understandable to have concerns, even with a supportive family.
What you're experiencing is fairly common. Despite having a good relationship with understanding parents, it can still feel daunting to open up about your sexual orientation to the ones you love. There may be fears about being judged, criticized, or having your identity invalidated.
A few thoughts that may help...
Give your family some time. Even if initially surprised, caring parents usually come around, especially when they see this is important to you. Be patient and keep communicating openly. It may also  be helpful to provide them with educational resources about bisexuality if needed. Sometimes people have misconceptions that information can clear up. Remind them that this is simply about you being your authentic self.
Loving partners of any gender doesn't change who you are as a person. Make it clear you aren't asking for permission to date, just their acceptance and support in being true to yourself. Your family may just need some time to process this new information before they can have an open discussion about it.
While you hope for acceptance from your loved ones, it's also wise to brace yourself for the possibility of negative reactions, at least initially. Even open-minded families can sometimes struggle when a child shares news that challenges their previous assumptions. If you do encounter rejection or hurtful comments, try not to internalize these messages and avoid lashing out if you can to avoid it. Instead, remind them calmly that your sexual orientation is simply a part of who you are --it doesn't change the amazing person they've known and loved all your life.
Don't back down on being your authentic self. You might find it helpful to reach out to LGBTQ+ support groups, understanding friends, or a counselor to practice having these conversations and process their outcomes in the aftermath. If resolution proves extremely difficult, you may need to prioritize surrounding yourself with your chosen family who can provide the affirmation you deserve. Stay patient, keep communicating with compassion, and above all, refuse to be shamed for courageously claiming your identity. With patience and honesty, your family should come to understand and embrace all aspects of who you are.
No matter what the outcome is, we are here for you. Thank you for this question.