From your post here is what I understand the issue is:
When I try to lean forward and try to increase my speed, my hands get fatigued and I can’t do it for longer durations....How can I go faster on a hybrid bike with this tyre and handlebar combination?
So, you'd like to find a way to be comfortable leaning forward and to go faster for a longer period of time.
In my experience leaning forward shifts more of your body weight to your hands.
Here is a description of the issue I think you are describing:
There are several nerves in your hand and if they are compressed then you will start to feel that tingling sensation. The ulnar nerve runs through the bottom of your wrist and to your pinky and ring finger, where as the median nerve runs through the middle of your wrist and to your thumb, index finger, middle finger and ring finger.
Here's what has worked for me to reduce hand fatigue.
- Riding position, your wrist and hand need to be in line with your forearm. From the article linked above:
To prevent numbness setting in you need to ensure that your wrist and hand position is in line with your forearm. If there is a bend in your wrist it will cause a pinch in the nerve and your hands will go numb. Per Swifty's suggestion below - rotating your brake levers to improve your wrist, hand, forearm alignment can help.
- Cycling gloves. Gloves will provide a layer of protection that spreads your weight over a larger surface area and offers cushion to absorb a little of the shock going into your hands from the handlebar.
- Use different hand holds. Find different ways to hold the handlebars to shift the pressure point of the handlebar to different parts of your hand. I'll shift my grip from a normal palm grip to a more finger-tip or a mid-finger-tip grip as an example.
- Different handlebar grips. There are many kinds grips designed to spread your weight over a larger area of your hand. Below is one example - not an endorsement just using this grip as an example.
- A combination of all these suggestions.
What the other comments/answers are saying is true. A hybrid bike is designed for on/off road use and has a relatively upright seating position. As you go faster wind resistance increases dramatically. If speed is your goal you will need a bike that puts you into an aerodynamic position.
According to Sheldon Brown:
At approximately 12 km/h, rolling and air resistance are equal. At higher speeds, air resistance is strongly dominant.
Here is a chart from Sheldon Brown's site to illustrate the challenge:
Without leaning forward your hybrid is something like the "Dutch Style Upright" bike in the graph.
Leaning forward helps but it still won't get you to the "Racing Bike" curve on the graph.
Note: Per Ian's suggestion below - "Gold Rush" is a recumbent bike
You should be able to find a way to prevent your hands from tingling. But, it takes a lot of work to make a hybrid go fast.