When the external voltage is applied, the depletion region gets thinner or fatter depending on the external voltage's magnitude and polarity.
When the p-n junction is unbiased, the p-type side is negative and the n-type side is positive, from thermal diffusion of carriers.
In the reverse-bias case, the n-type side is made more positive by the battery. The battery is doing work, shoving charges around to force the n-type side to be more positive and the p-type side to be more negative. The applied voltage forces negative charge carriers to the negative side and positive charge carriers to the positive side, so there's even more depletion in the charge-separated region, and conduction gets even harder in that region. The bigger the externally applied voltage, the wider the region with an electric field across it.
In the forward-bias case (negative terminal of battery attached to n-type side, positive side of battery attached to p-type side), now the external voltage is working against the diffusion-created p-n voltage. The n-type side, which was positive from diffusion, becomes now less positive; the p-type side, which was negative from diffusion, becomes less negative. The electric field in the depletion region becomes smaller, and the potential difference decreases (over a smaller depletion region). If the forward-bias potential difference increases yet more, eventually the depletion region disappears altogether, so there's no poorly-conducting region. The p-n junction effectively becomes a resistor, with a potential drop across it determined by the properties of the material. The majority (dopant) carriers can flow through the junction.