One great thing about these conversations is that they are upsetting. That may sound strange, but those kinds of discussions keep you awake at night thinking about them, which means you will get better. Next time you'll have better answers to those objections because you thought about it (maybe even obsessed about it). You gave it so much thought because it was upsetting. So don't fight it. Your feelings of upset are not pleasant, but they aren't a bad thing.
Several things will help you live with this kind of upset:
1. Take breaks in between. Don't have so many upsetting conversations that you aren't getting any sleep.
2. Take your time with the discussion. You don't need to respond immediately. You can take all the time you want, and then come back and say, "I've been thinking about our conversation and..." And you don't need to "win." You don't need to convince someone you are right. Just plant some solid facts, and let it go. They'll sink in and the person will have to alter the way they look at the world, even a little bit, to accommodate the new information. They might even look it up to see if you are correct, and when someone discovers the truth on their own, it is even more powerful.
3. Never overstate your case. Understatement is far more persuasive in the long run than overstatement. Try to be as accurate as possible, but if you're going to err, err toward understatement.
4. Don't spoil family gatherings with upsetting arguments. It's not worth it. And group settings are usually not the best venues for changing someone's mind anyway. If you have Islam on your mind at a family gathering and want to do something, build rapport. It'll help future conversations.
5. Think about the upsetting feelings differently. Rather than being a bad sign, think of them as a good sign that you're fighting for a cause that's important enough to get upset about.
6. Use your upset as fuel and focusing power, motivating you to look up facts and craft your arguments carefully. Use our Answers to Objections list to help you. Use the non-Muslim forum to request possible answers to an objection that has stumped you. As you obsess, take notes. It will help you remember the points you want to make.
7. Always try to give some fundamental information. Primary information can help change someone's world view in the long run. Muhammad killed people and took slaves. The Koran says 91 times that a Muslim should follow Muhammad's example. Simple facts like these are something solid that can help someone think differently about the problem of Islam.
These conversations are happening more frequently these days and with less resistance. But there are still plenty of holdouts who don't want it to be true and will fight you tooth and nail before they will admit (even to themselves) that the facts have overwhelmed their denial. This is a bitter pill to swallow, and some people will go through all five stages of resistance to the truth about Islam. Help them get through it and never give up.