What should I use over a water based polyurethane stain to make my project shiny and smooth?

Minwax polycrilic comes in several sheens. Have you tried the gloss variation? It should reflect the most amount of light.

There's some interesting words in here, huh? "Polyurethane stain" is an interesting combination. And what exactly is "varnish?"

Traditionally, stain is just about color. It's some dye or pigment suspended in a chemical that will easily evaporate. Just there to color the wood. Does nothing to protect it or add/subtract gloss.

A finish--either a lacquer or varnish--would be applied over a stain as a protective layer. Anything from shellac (excretion of the female lac beetle), tree resin, or polymers such as polyurethane are used for this. These protect the surface from physical damage as well as water/dirt/oil ingress, plus they build up that pretty transparent layer on top of the wood.

Rule of thumb: lacquers are just resins dissolved in a solvent, and when the solvent evaporates, it leaves behind the resin, meaning they can be repaired/modified/removed by applying more solvent. Varnishes undergo some chemical change during the drying process kind of like concrete; it doesn't just "dry," it "hardens."

Another rule of thumb: lacquers are often designed to dry very quickly, and are often designed to be sprayed on. Most commercially produced furniture has a lacquer finish simply because of how fast it can be applied; a table could be taken from raw wood to ready for the customer in under an hour. Varnishes are often designed to dry slower and are often designed to be wiped or brushed on. Varnish is more typical for floors, bar tops and other high wear applications because they're often more durable, but it can take days to apply in multiple layers.

Polyurethane is a varnish. To achieve a shine, you'll want to build it up in several layers, lightly sanding in between each. Water based polyurethane is easier to work with than oil-based, but it doesn't build up to the same gloss.

I suspect by "polyurethane stain" you mean you've got something like Minwax Polyshades, which is a urethane that has some pigment in it. These are marketed as "one-step" color and finish coatings. You would apply several coats of this product, lightly sanding between coats, and then apply a polish over it for your desired level of shine.

Mind you: A lot of things you might think of as "polishes" at first are actually varnishes or varnish-like materials. Linseed oil, for example, is a drying oil. It goes on as a liquid, but it reacts with the oxygen in the air and undergoes a chemical change into a solid. Danish oil, and Minwax's "Tung Oil Finish" are essentially thin urethanes. That latter product, the "Tung Oil Finish" is controversial because it has little to no tung oil in it; Minwax is trying to say it forms a tung oil-like finish. I personally like the product and use it a lot, and mentally think of it as "I Can't Believe It's Not Tung Oil." Applying these over a water-based urethane probably won't end well; one probably wouldn't adhere to the other and it would peel off or otherwise misbehave.

What you're probably going to reach for is paste wax. Paste wax is just a wax and oil mixture, lots of different oils and waxes have been used over the years. Once upon a time you'd find paraffin and turpentine used, which does a great job while being really poisonous. On the other hand, a popular mixture for food-safe applications is mineral oil and bee's wax, which is safe to eat by the spoonful if you're so inclined. It's a great polish for children's toys and furniture or wooden kitchen implements. Just wipe some all over the surface, let it dry for several minutes, then buff it off. Paste wax is also useful around the shop as a lubricant and rust preventer for metal surfaces such as the bed of your table saw.

TL;DR: Apply several layers of your water-based urethane, sanding between each, then polish with paste wax.

Danish oil is my go to when I'm in the UK, if it's available in your region, in France, I use libéron vernis bistro.