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As you're using Layout Managers you need to know that you can combine them. You might ask "How can I do that?"

Easy, nest JPanels inside each other, each with their own layout manager.

For your specific case, I see the following structure:

JFrame (The window)
 JPanel (The main pane) - Flow Layout (Default layout of a JPanel)
 JPanel (Left pane) - GridLayout (As your image shows, all labels & fields have the same size)
 JPanel (Right pane) - GridBagLayout (It's a weird accomodation of the buttons, so I did my best to replicate & give a better look)

As you can see, we have 2 nested JPanels inside another one and each of them has different layouts which help to have different accomodations.

The following code produces this output:

enter image description here

import java.awt.Color;
import java.awt.GridBagConstraints;
import java.awt.GridBagLayout;
import java.awt.GridLayout;
import java.awt.Insets;
import javax.swing.BorderFactory;
import javax.swing.JButton;
import javax.swing.JFrame;
import javax.swing.JLabel;
import javax.swing.JPanel;
import javax.swing.JTextField;
import javax.swing.SwingUtilities;
public class AdminForm {
 private static final String[] labels = {"Car Number", "Car No. Plate", "Model", "Capacity", "Date Purchased",
 "Insurance Status", "Date Insured", "Insurance Expiry Date", "Availability"};
 
 private JFrame frame;
 private JPanel pane;
 private JPanel leftPane;
 private JPanel rightPane;
 private JTextField[] fields;
 private JButton searchButton;
 private JTextField searchField;
 private JButton updateButton;
 private JButton deleteButton;
 private JButton clearButton;
 private JButton nextButton;
 private JButton previousButton;
 
 public static void main(String[] args) {
 SwingUtilities.invokeLater(new AdminForm()::createAndShowGUI);
 }
 
 private void createAndShowGUI() {
 frame = new JFrame();
 pane = new JPanel();
 leftPane = createLeftPane();
 rightPane = createRightPane();
 
 pane.add(leftPane);
 pane.add(rightPane);
 
 frame.add(pane);
 frame.pack();
 frame.setVisible(true);
 frame.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
 }
 
 private JPanel createRightPane() {
 JPanel panel = new JPanel();
 
 searchButton = new JButton("Search");
 searchField = new JTextField(10);
 updateButton = new JButton("Update");
 deleteButton = new JButton("Delete");
 clearButton = new JButton("Clear");
 nextButton = new JButton("Next");
 previousButton = new JButton("Previous");
 
 panel.setLayout(new GridBagLayout());
 
 GridBagConstraints gbc = new GridBagConstraints();
 
 gbc.gridx = 0;
 gbc.gridy = 0;
 gbc.fill = GridBagConstraints.BOTH;
 gbc.insets = new Insets(5, 5, 5, 5);
 
 panel.add(searchButton, gbc);
 
 gbc.gridy++;
 gbc.gridwidth = 2;
 
 panel.add(searchField, gbc);
 
 gbc.gridy++;
 gbc.gridwidth = 1;
 
 panel.add(updateButton, gbc);
 
 gbc.gridy++;
 
 panel.add(deleteButton, gbc);
 
 gbc.gridx++;
 panel.add(clearButton, gbc);
 
 gbc.gridx = 0;
 gbc.gridy++;
 
 panel.add(nextButton, gbc);
 
 gbc.gridy++;
 panel.add(previousButton, gbc);
 
 panel.setBorder(BorderFactory.createLineBorder(Color.GRAY));
 
 return panel;
 }
 
 private JPanel createLeftPane() {
 JPanel panel = new JPanel();
 fields = new JTextField[labels.length];
 
 panel.setLayout(new GridLayout(0, 2, 5, 5));
 
 for (int i = 0; i < fields.length; i++) {
 fields[i] = new JTextField(10);
 
 //This makes text fields not editable for certain ones.
 switch (i) {
 case 4:
 case 6:
 case 7:
 case 8:
 fields[i].setEditable(false);
 break;
 default:
 break;
 }
 
 panel.add(new JLabel(labels[i]));
 panel.add(fields[i]);
 }
 
 //Investigate how to change color of border, etc
 panel.setBorder(BorderFactory.createTitledBorder(BorderFactory.createLineBorder(Color.GRAY), "Update Data"));
 
 return panel;
 }
}

Just play around with the values to have your desired output.

Side note:

Give your variables better namings, later on you'll ask "What is f"?

JFrame f = new JFrame("Panel Example");