Help a newb get a usable web based email set up pleeease!

Hello, I have email online through time warner cable and it is soooo slow, taking five to thirty seconds to delete mail, advance pages, etc. I need a free web based mail client that will work with my existing roadrunner email (which is my primary email. I hardly ever use the gmail account) quicker than that. I tried going to outlook mail on line and adding my account but it gives a constant update in progress message for more than 24 hours so I don't think that's going to work. I don't have any windows mail program installed on my computer itself. I've tried Thunderbird and it constantly had issues with forgetting the password and even after I'd type it in it would say it wasnt correct, so Thunderbird is not an option for me either. I don't want to give up or switch my email address again! Please help. I'm just not tech savvy and up to date with a lot of this info. I don't even know if microsoft offers a free web based email. I'm just hoping. Thanks.