I think you are smart to not give them your number right away. I know that I would not feel comfortable giving out my phone number to a stranger, especially within the first few messages.

That being said, as Anya mentioned, some men want to know if you are "real" or not. For example, they might think you are employed by the website or an agency and are trying to scam them in some way. Asking and receiving your phone number is a way to "verify" you, because most scammers would not give out personal details.

A good alternative is to provide your email address and suggest communicating that way. Tell the men that you are not comfortable giving out your phone number so soon but would be happy to correspond through email. This way the men feel like you are interested in them and willing to talk through something that's a little more personal than messages on the site. Plus it will remove most of their concerns that you are employed by the website because you are willing to move communication away from it. If you don't want to actually use your personal email you can just make another email address to use. Once you feel comfortable enough then you can move on from email to phone numbers, Skype, Viber, etc.

I think most reasonable men with serious intentions would understand this. If they insist that you provide your phone number (or address) right away and will not accept anything less, I would move on and find somebody that is more understanding.