Great question! Turboprops spin way slower so even though they’re larger the resulting tip speed is slower.

E.g., turbo prop (source https://www.quora.com/How-fast-is-the-tip-of-a-turboprop-bla...)

I double checked this math and got 474mph.

Meanwhile a Piper Arrow (4-seater piston, pretty comparable numbers to most Lycoming/Continental singles) takes off at 2700 rpm and has a 74” propeller, for a tip speed of 594mph. You can use a 3-blade propeller to reduce the diameter and cut noise somewhat (among other benefits).

Both of these are just the tip speed relative to the plane, I suppose you’d need to factor the airspeed too and use Pythagorean theorem but I’m no aerospace engineer.

As you noted, this is possible because the constant-speed prop on the turboprop allows it to take larger bites of air.

The Arrow needs that rpm to produce full takeoff power. However after takeoff it can be adjusted to a lower speed with minimal loss in power (e.g., 2500 rpm greatly reduces noise while losing 7-8% power). To reduce RPM like that in a fixed pitch you’d also need to reduce manifold pressure, causing more power loss.