Sounds like your carburetor isn't up to the task. That, or the fuel pump is shot.
After it conks out, look down the carb throat and work the throttle manually. Do you see squirts of gas every time you goose the throttle by hand? If not, your accelerator pump is bad.
Hook a hose up to the carburetor and put it in a gasoline can or something and let gravity feed the carburetor. See if it runs, then. If you have an electric pump , try that.
