Two answers:

Easy (possible) solution: You should be able to take off your fan cover plate and possibly unplug your fan motor unit (power off to fan!) and remove it from the housing. Take it to a home store and find another fan motor that will fit back in that housing. With any luck, you can have a brand new fan with a little more effort than changing a light bulb.

Long Term Solution: It sounds like you lack proper insulation in your walls and/or ceiling. If that is the case, you will always have a potential mold problem since the moisture will quickly condense on the cold surfaces. A mold-resistant paint and primer might help, but the true solution is to re-insulate. See the linked page below.

Here is a link that might be useful: How To Eliminate Bathroom Mold

There are a number of things you should do before heading out to the local home supply store with credit card in hand.

1. Clean the existing fan. It must be remembered that the fan is discharging moisture laden air therefore it stands that the fan blades or squirrel cage will get wet in the process. It further stands that as the air in the bathroom passes through those wet blades a large percentage of the dust in the air will accumulate on the fan in the form of a mud. That mud will then dry in place and the process continues until the efficiency of the fan is reduced to a minor fraction of its designed capability. The solution is to simply periodically clean the fan.

2. Check the draft damper- nearly all exhaust fans have a small sheet metal damper on the air exhaust port that is hinged from the top and remains down in the closed position by the mere physical weight of the damper. When the fan is running the air forces the damper open allowing the air to discharge. here again, during the running cycle the damper gets wet and the dust accumulates adding weight to the damper and interfering with the hinging action. Solution- Periodically clean the damper when cleaning the fan.

3. Check the fans exhaust duct. Quite often we find that the fan is connected to the exterior discharge port via a flexible duct and sometimes when ppl are working in the attic space that duct gets shoved aside causing a kink in the line, or the exterior port may be obstructed by a birds nest or wasp nest.

4.Remember that nature abhors a vacuum. No matter how large or how efficient your fan is, it simply cannot exhaust air unless there is a provision for an equal amount of air to enter the room. Try a simple experiment. After you clean the fan close the bathroom door and listen to the fan, then open the door. If you hear a substantial change in pitch of the sound it is indicating that there is not enough make up air entering the room. Generally in residential construction make up air enters via the crack under the door or though the HVAC ducting, however in some instances it is necessary to install an air transfer duct through the wall to the next room with a trim grille on each side.

5. Make sure the fan you have it properly sized for the room and not just the cheapest one they could find to meet code when it was installed.

Properly the exhaust fan should provide 4 complete air exchanges per hour. Fans are rated in CFM and the rating should be listed on the units data plate.

To compute your required size begin by determining how many cubic feet of space is in the room. To do so, multiply length x width x height.

Example, let us assume a room 10'L x 8'w with an 8' ceiling. 10 x 8 x 8= 72cubic feet.

We want 4 complete air exchangers per hour so we multiply the total volume x 4 to determine how many cubic feet of air we must move per hour.

From the example 72cu.ft x 4 =288cu.ft.

There are 60 minutes in an hour so we divide the total load per hour by 60 to determine the CFM (cubic feet per minute).

288/60= 4.8CFM

6. Fan timer- Remember that at 4 air exchanges per hour it takes the fan 15 minutes to discharge the moisture laden air. If your fan is controlled by a simple on/off switch when you turn the light and fan off as you exit the room any moisture in the room will remain. The solution is to replace the fan control on/off switch with a 15 minute delay on break fan switch. in this manner as you turn the fan and lights off as you exit the room the fan will continue to run for an additional 15 minutes then it will shut off automatically.

Thanks lazypup, you have changed my frame of thinking about this.I will begin to inspect and clean my fan and make sure it is working as it should.Installing a louvered vent to an adjacent room would help to create more airflow.thanks for the tips,I will let you know what i find.

good info lazypup!
note #2
some bath fans have plastic back draft dampers
that connect to the housing of the bath fan.
make sure that yours is installed..installed
correctly (closed position when not in use)
and seal it to the bath fan housing
and that it is sealed to the venting.
just wanted you to get in the attic..LOL!
best of luck.

O.k I cleaned the fan and the damper is working,there was alot of dust and mold on it.There still seems to be alot of moisture left after a shower is taken.I can see water streaks on the walls after it has dried.We usually take our showers in the morning temp is around 65 - 68 Deg. I have the thermostat set down lower at night.
I am now thinking of installing a vent that simply goes into the adjacent bedroom because there is no register in the bathroom.Maybe this would help with the airflow better.

Hi, sounds like you are in a cold weather climate so I would not go with a window.
An experiment you can try is next time you shower make sure the door is closed and when you're done look at the bathroom mirror probably covered with mosture.Then the next time leave the door open when you're done check the mirror if it is mostly clear I would upgrade the fan and vent the room. I have hung louvered door in bathrooms to beat this problem.
Good Luck Woodbutcher

10x8x8 does not equal 72 cubic feet.

You know the towel that you just dried yourself off with?
Wipe the walls with it.
Leave the fan on while you are dressing, turn it off as you walk by on you way out.