Leg strength isn't a big factor for going fast on the bike. You can see from the chart below that anything over a couple minutes is primarily aerobic. It's the reason guys with legs no thicker than pencils can win the Tour de France
If your quads are tiring maybe try increasing your cadence a bit. Typically the faster I go, the higher my cadence. A common beginner mistake is to slog away in a gear that's too big. There's no "ideal" cadence but try finding something between 80-100rpm that works for you.
A good dose of miles 4x or 5x/week with varying speed done consistently over time with appropriate recovery is the main thing to getting faster. A few weeks isn't much, but keep at it regularly for a couple months riding an hour or two a day and you will start to see some improvement. Adding a longer ride of 2-3 hours on the weekend (or whenever) will go a long way to improvement. You should be getting your heart rate above 70% of your maxHR for most of the ride, and to maximize your potential should spend some of that time close to or even above 90% of your maxHR, although for a beginner that shouldn't be the priority unless of course you just enjoy it. A few minutes at that level of intensity sprinkled into a moderately paced ride goes a long way.
Once you have some basic fitness ( a couple months of daily rides), a more advanced strategy to increase cruising speed are intervals. One commonly used type is known as
SST (sweet spot training). Sweet spot because it's enough intensity to bring about physiological adaptations, but not too intense as to exhaust one so regular riding is interrupted, which is enemy #1 toward making progress.
How to do it: Warm up at an endurance pace (65%-70% maxHR) for 15-20 minutes, ride fairly hard for 15 minutes (about 80% maxHR), ride easy for 10 minutes, then repeat a second time. If you start to fade out toward the end, adjust the next time and go a little easier. The key is maintaining a steady effort across both of the intervals. Finish up by riding easy the last 10 minutes to cool down. The workout can be done in an hour and many guys on the forum have seen nice gains doing these workouts mixed in a couple times a week with their other riding. Many serious riders do 20 minute variations of these intervals (you'll often see this referred to as 2x20), but you can easily make progress starting out with a couple 15 minute efforts.
Eating something before a ride is good... imagine trying to drive a car without gas in the tank.