Hi and welcome to the Forum!
Well you're doing all the right steps. I'm assuming you're on OS7. If you are copying a web address from the URL itself, just tap the URL, the whole address will highlight blue, the screen changes, just tape the blue again and the Copy/Paste box will appear, just press Copy and then cancel out of the screen (you'll find Cancel to the right of the URL line). If on an earlier OS just tap the URL, this will bring up the keyboard, tap the URL again and it should bring up the Select/Select All tab, press Select All then Copy from the next option box.
If trying to copy from within the screen page, most website addresses will be an actual link, just press and hold on the link and you will get an option box to Open, Open in New Tab etc.....one option will be to Copy.
If this isn't working for you, the app may be playing up......try closing it completely and then relaunching the app. Let us know how you go.
From the home screen do the four finger swipe up or double press the home button to reveal your multi task bar. These are all the apps that you have opened at one time. If you have updated to IOS7 - just swipe up on the image above the app icon to close it. If on an earlier OS - Long press the app until it jiggles and a red minus sign will appear above it. Press the minus sign, this will close the app, not delete it, now press the home button to return to your screen.