Don't be deceived by what your computer calculates as the average speed. As MichaelW's response suggests, there are lots of external factors (like stop lights) which affect it. So, use it only as a broad benchmark.

Rather, figure out what kind of typical speed you want to maintain on the flats or on hills of various grades (as reported by your cyclocomputer as you are riding), and use those numbers as benchmarks.

But, like the others have said, intervals is the way to go. Ya gotta just get stronger. And be patient. If you rush it you'll hurt yourself one way or another.

Cheers,
Jamie