Hi, jackiejanep!
Yes, it is normal as far as anything can be normal in sweeping. So much is random, and there is so much variation with the kinds of sweeps people enter, that it is hard to compare things between one sweeper and another...but, often in the forum, people say it generally takes about 4 months before you start winning. That is based on things like how long it takes to get started (and go through the lists of existing sweeps) how long sweeps run, how long between the end date and winner selection/notification or the time between winning and prizes being sent (often 6-8 weeks after the end date of the sweeps.) As with anything, there are exceptions - a handful of people win something big right away.
Have you read amyshulk's Sweeper's Credo? Or the thread, Not Winning Anything? Tips on Why? I will look for the links and post them here. Main things: don't give up, enter and forget it, be persistent, and if you are not enjoying the hobby, take a break...