In a word, “yes”, so long, of course, as proper technique is used. Rest assured, the belief that weightlifting might stunt growth is completely false. As children age, their bones lengthen. Epiphyseal plates, otherwise known as growth plates, are located in between the bones. When signaled by the brain through hormonal processes, these plates extend the bones throughout childhood, culminating with the completion of puberty, at which point the growth plates are closed, and the young person’s height is set. As such, height growth is purely hormonal, and the only way strength training could affect it is by way of causing a bone to break, which could adversely affect its growth plate. Luckily, this is almost impossible, and far more likely to happen from more popular sports.
Health care and fitness professional groups, including the American Academy of Pediatrics, the American College of Sports Medicine, the American Orthopaedic Society for Sports Medicine, and the National Strength and Conditioning Association, agree that a supervised strength training program that follows the recommended guidelines and precautions is safe and effective for children [1-5]. When well supervised, such a program has no greater inherent risk than that of any other youth sport or activity. 
If a child is interested in strength training, it should be encouraged. Children of all ages can safely exercise in the weight room given proper supervision and an appropriately individualized training program for the child. Proper exercise progression should be used, and there’s little benefit to having a program with high volume, high frequency, or sets taken to failure. Proper technique should be emphasized, adherence to which should be the primary goal of any child-oriented program.
Weightlifting has many benefits for health and wellness, such as improved coordination, strength, bone density, and even cardiovascular fitness. Those are some great reasons to ensure that children enjoy the strength training they’re exposed to, which can in turn ensure that they reap its benefits for many decades to come.
There’s no minimum age required, but around 7 or 8 years of age is a good guideline, assuming the child is interested. A simple routine consisting of 1 to 2 sets of compound movements (such as goblet squats, rows and presses) 2 times per week can be utilized. While this is a good starting point, the program should, of course, be adjusted, depending on individual needs and preferences. It’s worth noting that children who want to improve sports performance would likely see a greater benefit from just focusing on practicing specific skills of their sport, though doing so is certainly not mutually exclusive to strength training: if such children are willing and able to do both, then more power to them (pun intended)!
In a world where childhood obesity and sedentary lifestyles are all too common, strength training should be welcomed, not discouraged on the basis of unevidenced myths.
1. American Academy of Pediatrics Committee on Sports Medicine and Fitness Strength training for children and adolescents. Pediatrics. 2008;121:835-840
2. American College of Sports Medicine Current Comment: Youth Strength Training. Indianapolis, IN: American College of Sports Medicine; 1998
3. Cahill BR. American Orthopaedic Society for Sports Medicine: Proceedings of the Conference on Strength Training and the Prepubescent Chicago, IL: American Orthopaedic Society for Sports Medicine; 1998
4. Faigenbaum AD, Kraemer WJ, Cahill, et al. Youth resistance training: Position statement paper and literature review. J Strength Cond Res. 1996;18:62
5. Schafer J. Prepubescent and adolescent weight training: is it safe? Is it beneficial? J Strength Cond Res. 1991:13:39
6. Faigenbaum AD, Kraemer WJ, Cahill, et al. Youth resistance training: Position statement paper and literature review. J Strength Cond Res. 1996;18:62