How do I increase my average speed?

• Being in the correct position will allow you to use your legs more efficiently. Raising the seat will help. If you're sharing the bike, make marks on the seatpost so you can quickly and consistently change between positions.
• Aerodynamics are very important in cycling, and the faster you go, the more important they are. There's not a lot you can do to change your aerodynamics on this bike—hybrids put you in an un-aerodynamic position.
• As you might imagine, getting stronger as a rider will make you faster, so keep riding.
• Turns don't lower your average speed that much. I wouldn't worry about that.

First of all I’d fix the saddle position. A low saddle will decrease your power output and can cause knee problems. As a rule of thumb your leg should be fully extended with your heel on the pedal in the 6 o'clock position.

Keeping the handlebars low or even lowering them further will generally improve your aerodynamics.

Make sure the tires are inflated to a relatively high pressure (depending on your weight and road surface, probably around 4–5bars) to reduce rolling resistance. Make sure the chain is lubed properly to reduce friction.

Wear tight fitting clothes to improve aerodynamics.

Regarding turns: Here I disagree with Adam Rice, a lot of turns will certainly decrease your average speed, especially if you have to brake. Good brakes and technique can allow you to brake stronger and therefore later, thus improving your average speed.

Since it sounds like you are using average speed as a benchmark of fitness or power: It’s relatively useless for that, even if you only use it to measure your own progress. Even on the same roads different wind or traffic conditions can cause a lot of variation.

The easiest thing you can do to make a hybrid faster is change the tires. Narrower, higher pressure tires with no tread roll faster (on flat, smooth surfaces). If you already have relatively narrow tires (32mm) with minimal tread, inflate them to the maximum pressure.

If the roads in your area are rough, wider tires at lower pressure will be needed.

Regarding turns, stops at junctions etc. Strava/bike computers will pause if you stop and will give you an average moving speed. You can compare average speed for rides on similar roads with a similar amount of stops. When you are making frequent stops average speed will reflect not just your general moving speed but also your ability to accelerate back up to that speed after stopping or slowing down.

Based on the assertion you do not want to change the saddle height, aerodynamics remain the single biggest factor in speed improvement on flat ground that you have available to you. A good graph here shows the effect. If you are riding at 30km/h in an aerodynamic position, doing nothing except getting into an aerodynamic position will increase you speed to 35km/h - not a bad gain for free*. Above about 15km/h aerodynamic drag has a bigger effect than is the biggest item in the equation.

To improve your aerodynamics tuck elbows in, move hands inward on the bars (Maybe move the grips and brakes/shifters inboard to make if more permanent), tuck in low to the bars. Only so much you can do and be comfortable on flat bars, and too much will loose power. Experiment with various degrees of tucked in positions to find one that is comfortable and efficient and safe (e.g. can you get to the brakes if needed). You could go to the extreme and bolt on some aerobars.

Already mentioned - raise the saddle to the correct height. Ignoring your 'No parts' request, a quick release on the seat post so you can quickly and easily adjust the seat height between what you and your father like s would be a very valuable investment.

Lowering the bars would make a small difference, but given its a shared bike (and not yours), I would not recommend it until you have done other things.