Great question... and an important point about glucose and lipid utilization! The body can't turn fat into sugar, but it can turn sugar into fat. The more technical way to say this is that the body can't convert acetyl-CoA into glucose, but it can convert glucose into acetyl-CoA.
The enzyme that converts glucose to acetyl-CoA, pyruvate dehydrogenase, catalyzes an irreversible chemical reaction. Pyruvate dehydrogenase converts pyruvate into acetyl-CoA, which can be used in the Krebs Cycle, fatty acid synthesis, etc. Once acetyl-CoA is produced it cannot be converted back into glucose (carbohydrate) within the body. This is important because certain cells/tissues can use only glucose for energy production, while other cells/tissues like the liver can use both glucose and lipid for energy production.
This is where your question becomes very important, because the liver is the predominate organ that maintains glucose homeostasis (i.e. maintains blood glucose concentration) in-between meals or during periods of fasting. In periods of fasting, the liver uses free fatty acids released by adipose tissue for energy/ATP production in order to produce glucose through gluconeogenesis, an anabolic process that requires energy.
So, growth hormone increases fatty acid release from adipose tissue stores for uptake by the liver. The liver takes up the fatty acids from the plasma and uses them to produce glucose for release into the circulation for glucose-dependent tissues to use. During this time of fasting, tissues that can use fatty acids and glucose tend to switch to fatty acids as their energy source, while tissues that only can use glucose will continue to rely on glucose production from the liver for their energy source.
Here is a nice summary (open source) from National Library of Medicine explaining in more depth the basis for why fatty acids are oxidized to provide ATP for gluconeogenesis - "Energy Metabolism in the Liver".
Also, George Cahill, a famous physician-scientist at Harvard University who studied metabolism and starvation has an excellent review of fuel flux under fasting and starvation conditions that demonstrates the above mentioned principles as well "Fuel Metabolism in Starvation" - namely that free fatty acids released from adipose tissue are oxidized in the liver and used for anabolic processes in the liver, such as gluconeogenesis that maintains blood glucose concentrations.
