Can a small displacement engine with bigger turbo or compressor beat bigger displacement engine

You are in the wrong forum for this. Chrysler built 2.5L four-cylinder turbocharged engines that developed over 500 horsepower from a stock engine block to prove what their lower end was capable of. The jump from 400 to 500 horsepower was achieved with nitrous. That does not mean it was suitable for daily driving on the road. The fact you have to ask this question tells me you know even less about racing engines than I do, and you are in way over your head.

The first issue is if you research any engine that comes from the manufacturer with a turbo, you will find it has a lower compression ratio than standard engines. They will be on the order of 7.5 / 1 instead of 8.2 / 1. A higher compression ratio causes an engine to develop more power from a given amount of fuel, but at a trade-off for other factors. Lower compression engines are very low on power, but that is made up by adding the turbo.

You will also find that any turbocharger kit you can buy comes with the disclaimer that it is intended for off-road use only. The manufacturers of those kits know they will not provide satisfactory performance for street cars, but that doesn't stop them from selling them to unsuspecting customers.

A turbocharger kit includes everything you need to make the engine develop more horsepower. A turbocharger option that came on a new car includes everything needed to make the car last. You will get bigger brakes, rotors, and tires, a larger radiator, stiffer suspension parts, stronger front coil springs, a brake system proportioning valve calibrated for the car's weight distribution, an external oil cooler, and often an inter-cooler. Are you prepared to add all that stuff to your car?

You also have to think about "torque steer". That is when the car veers unexpectedly to one side during hard acceleration. Chrysler eliminated that a long time ago, and other manufacturers made the same suspension system modifications to make it possible to control their cars. How are you going to handle torque steer on your car? You might want to find a Dodge Omni GLH and drive that to see what torque steer is all about. Be sure your life insurance is paid up!

Keep in mind that most crankshafts and connecting rods, and their bearings, are designed to handle the amount of power the engine is expected to produce, and not much more. Internal parts need to be built stronger to hold up to higher power. That is why a lot of engines are destroyed by adding a turbocharger. Before you give this idea any more thought, visit any engine machine shop or race engine shop and run your idea by them. I have the owner of just such a shop in my list of contacts who I share political cartoons and jokes with. I have had this discussion with him in his shop before, and once the laughter winds down, it always ends up with the same comment: Your car will appear in your front yard with a "For Sale As Is" sign in the window. Use your vehicle for what it was meant for. If you think you need a turbocharger, buy a car that came with it from the factory. If you want a major modification that neither of us is emotionally, physically, or intellectually prepared to perform, hire an automotive speed shop, and have them tell you what is involved. If you think you are building a car to race, at least start with one that is strong enough to hold up the weight of the roll cage. I am sorry to dash your dreams, but it would be nice if you were still around years from now to come back here when you need us.

You are welcome. Look at this as an opportunity to trade your vehicle for what you want. The manufacturers build much nicer packages than we can add on later.