Yes, "tilting"/banking the aircraft could cause a reduction in the upwards component of lift if nothing was done to compensate. However what usually happens is that the total lift produced by the wing is increased by increasing the wing's angle of attack - simply put the pilot will pull back ever so slightly on the controls ( you may perceive this in the cabin as a slight increase in "g"..end result of all this is the upwards component of lift - the component opposing gravity - remains the same and the aircraft doesn't fall to Earth. As an extreme example you can fly a 60 degree banked turn and not "fall" as long as the total lift is increased to double the value needed for straight flight = you would be pulling 2g..well above anything you would expect to experience in an airliner
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As others have said airlines have strict rules on bank angles ( typically no more than 25 -30 degrees) and don't allow turns on take off until well above the ground ( again typically 400 feet or more). Most airlines monitor compliance with these rules and a whole host of others through the Flight Data Recorders - see Mike Jenvey's comment.