Awkward copyright/release situation

I'm excited because I got my first "gig" photographing a horse show this weekend. Since it's my first show and I have a new camera, I offered to do some photography free.

However, the situation with the copyrights and all is kind've confusing...the organizer wants full control over the photos, in a way.

They say that they need me to "get a release for any photos you promote on your website and use in your 'portfolio'"
Sure, I understand and stated to them that I'll only take photos of riders who have signed releases, want their picture taken, etc etc. Completely reasonable.
But, I'm feeling like they're my photos. I most likely won't use them for commercial purposes (although it might be good to use on a business card or website layout, and I'd ask them before I did that) but I feel that I should maintain otherwise full control over them and be able to use them in my portfolio.
It's one thing to tell someone they can't use the photos they've taken for commercial purposes, I understand that, but to try and get extra control in this way over them?

I don't know. What do you all think? I'm new to the business side of photography and don't want to sign any restraining contracts for my photos. Any idea/thoughts as to how I should proceed? Thanks!

The law is different state by state (or country). In IL, you can use an image for your portfolio as a professional photographer without a model release. Other states - not. - TF

So this is a private event? If that's the case, then you really do need to abide by the rules the promoter demands - especially if its on private property. I don't know for sure, bur I expect OldClicker is right about the usage for portfolio laws differing from state to state. However, you really aren't in the driver's seat in this situation. If you want the "job," you have to take it as they offer it. Will the promoter give you contact info for the competitors so that you can ask them for releases?