In electrostatic shielding, we have read that the charged particles in a conductor experience no force. In the current carrying conductor, electric field is applied across its ends and the electrons experience force due to this field and constitute electric current. However the two ideas oppose each other. If there is no electric field inside conductor, then how there will be flow of charge?
The electric field is zero in a conductor only in electrostatic situations. And this happens after a (usually extremely short) time comparable to the so-called dielectric relaxation time $\tau=\epsilon/\sigma$ for the redistribution of charges. If you apply a steady voltage difference to a conductor, there will be an electric field inside the conductor which produces a stationary current.
The electric field inside an ideal conductor will be zero when under static conditions. In this case, a current can flow without an electric field present inside the conductor. But this is only true for an ideal conductor, like a superconductor. Most people, when talking about electromagnetics, will make the assumption that there is no electric field inside a conductor, however people talking about circuits will say that there is a (small) electric field inside the conductor.
In a non-ideal conductor, such as a real world wire or metal plate, a small electric field is required to maintain a constant flow of current under constant potential. This is only true because the electric field, which is directly related to the electric force on the charges in the wire, must overcome the non-zero resistance of the wire.
The mechanism at play in a typical circuit that allows for a constant potential is the existence of a power source which can maintain a difference in charge between two points, thus providing a constant voltage. In order for this to work, charges traveling into one end must be transferred to the other end continuously, or else the wire would "succeed" in rearranging all of its charges. This is done in any number of ways, e.g. electrochemically with a battery.
Side note: Power sources can also supply constant current, generating whatever potential difference is required to maintain a constant current. These are called current sources, but they are only common when you work with circuit design; they aren't as familiar so they are often left out.
It's worth noting that applying a potential difference to a wire will be effectively "shorting" the power supply, allowing very high currents. In the case of an ideal conductor, it would require infinite current to maintain a constant potential. This is why we need current-limiting elements such as resistors. Charges inside an ideal conductor will immediately rearrange, and the only way to maintain a potential is by allowing infinite charges through the power source, which is, of course, impossible.