Most likely a loose connection in the new panel or upstream.
When lights throughout the house get dim, are there a few that don't dim? Put an incandescent (old fashioned bulb) light on a circuit that does not dim for a second opinion.
Better yet, get a voltmeter (or multimeter) so you can measure exactly how many volts' drop you are experiencing. A drop of 2 volts can be ignored but a drop of fifteen volts is a bad situation. If incandescent lights dim very noticeably that is a bad situation.
This needs attention quickly. Allowing a loose connection to persist can result in damaging the panel or meter box or other location where the loose connection is. Turn off as many things as you can to reduce the total load until you can find and fix the problem.
Stand with both feet on dry cardboard, put one hand in your pocket, and carefully press your other thumb on non-metal parts including in the breaker panel. Unusual heat might be the location of a bad connection.
Not a too small feeder because if the feeder can't handle one hair dryer or one circular saw then the feeder won't handle the rest of the things in your house used several at a time. (Normally a feeder will not allow you to run absolutely everything all at the same time.)
Not a bad ground. If improving the ground improves the performance of electrical things that means something else, probably the neutral, is also bad. The neutral is part of the normal path, or circuit, for the electrical current. The ground is not intended to carry current but sometimes it does carry some.
Probably not a bad feeder wire. It is rare for a wire to break in the middle of a run so as to leave two hidden broken ends that make poor contact or to leave a thin section (of a few remaining good strands) in the middle.