What mechanism turns elliptical orbits into circular ones?

A tidally locked moon has tidal bulges. At perigee, when the Moon is closest and moving fastest, its rotation lags behind the orbital motion. The Moon is not rotating fast enough to keep its tidal bulges aligned with the planet. The nearside bulge is slightly ahead, the farside is behind.

Since the two bulges are at different distances from the Earth, the pull on the nearside one, which is leading, is stronger. But the pull is toward the center of the Earth, so there is a small force backward on the bulge, which decelerates the moon.

That effect of deceleration at perigee will lower the apogee, and therefore circularize the orbit.

Note that at apogee the situation is inverted, but the forces involved are smaller as the Moon is further away. So each orbit there is a net circularization.

Your question just about answers itself. The satellite gets a tidal bulge. Since the satellite is stiff to some degree (even if it is a gas giant), the bulge rotates with the satellite. But since the orbit is not circular, the bulge gets away from the gravity center line to the primary. So the satellite is pulled a bit backwards (or forwards) in its orbit. Over millions of years, this extra pulling tends to make the orbit more circular.

Google "circularization of orbit tidal heating" for more information.

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There isn't any mechanism that turns elliptical orbits into circular ones. There isn't such a thing as a true circular orbit occurring anywhere in practice.

ALL orbits are elliptical, even the best launched artificial satellites have some slight eccentricity.

If you see a table of data which indicates orbital data but doesn't indicate evidence of eccentricity or a value of zero, all that means is we haven't been able to measure that eccentricity, or either that or it isn't important to keep track of it.